Showing 1–50 of 190 results for author: Karniadakis, G E

Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces

Authors: Khemraj Shukla, George Em Karniadakis

Abstract: Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a p… ▽ More Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces. △ Less

Submitted 5 August, 2026; originally announced August 2026.

MSC Class: 68T07 (Primary); 41A65; 46A03; 65N30; 76D05 (Secondary) ACM Class: I.2.6; G.1.8; G.1.2

Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers

Authors: Jaemin Oh, Youngkyu Lee, Jerome Darbon, George Em Karniadakis

Abstract: Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative \(L^2\) error \(O(10^{-3})\) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared traini… ▽ More Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction. We show that this premise is unreliable. An operator trained to the relative \(L^2\) error \(O(10^{-3})\) can still produce an initial state in which the discrete Jacobian is indefinite, because the mean-squared training controls error on average while leaving localized pointwise violations of the underlying physics. For a nearly incompressible hyperelasticity problem, we trace this to the predicted volume change: the operator disperses \(\mathrm{det} F\) well away from one, and the resulting Jacobian acquires negative eigenvalues even when the predicted field is visually indistinguishable from the reference. At a small scale, this is a nuisance; at a multi-million degree-of-freedom scale, it is disqualifying, since the conjugate gradient and other Krylov solvers needed for memory-feasible Newton steps assume a definite spectrum. We then show that a short, label-free fine-tuning phase -- penalizing the operator against the discrete energy, with no additional solution data -- shifts the Jacobian spectrum back to positive definite. Combined with an inexact outer loop, this gives a warm-started Newton method that converges across the full loading range where the unregularized operator fails, reaching up to 5.4\(\times\) wall-clock speedup over incremental continuation on a 3D problem with 6.4 million degrees of freedom. △ Less

Submitted 18 August, 2026; v1 submitted 19 June, 2026; originally announced June 2026.

Comments: 23 pages, 8 figures, 7 tables

MSC Class: 65N22; 65H20; 68T07

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

Authors: Zhen Zhang, Alessandro Alla, George Em Karniadakis

Abstract: Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, op… ▽ More Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost. △ Less

Submitted 10 June, 2026; originally announced June 2026.

Comments: 35 pages, 10 figures

Agents' Last Exam

Authors: Yiyou Sun, Xinyang Han, Weichen Zhang, Yuanbo Pang, Tianyu Wang, Yuhan Cao, Yixiao Huang, Chris Duroiu, Haoyun Zhang, Jeffrey Lin, Weishu Zhang, Tyler Zeng, Ying Yan, Bo Liu, Hanson Wen, Mingyang Xu, Xiaoyuan Liu, Zimeng Chen, Weiyan Shi, Amanda Dsouza, Vincent Sunn Chen, Patrick Bryant, Carl Boettiger, Yamini Rangan, Bradley Rothenberg , et al. (285 additional authors not shown)

Abstract: Recent AI systems have achieved strong results on a wide range of benchmarks, yet these gains have not translated into economically meaningful deployment across many professional domains. We argue that this gap is largely an evaluation problem: widely used benchmarks lack sustained performance measurement on real and economically valuable workflows. This paper introduces Agents' Last Exam (ALE), a… ▽ More Recent AI systems have achieved strong results on a wide range of benchmarks, yet these gains have not translated into economically meaningful deployment across many professional domains. We argue that this gap is largely an evaluation problem: widely used benchmarks lack sustained performance measurement on real and economically valuable workflows. This paper introduces Agents' Last Exam (ALE), a benchmark designed to evaluate AI agents on long horizon, economically valuable, real world tasks with verifiable outcomes. Developed in collaboration with 250+ industry experts, ALE covers non-physical industries defined with reference to O*NET / SOC 2018 (the U.S. federal occupational taxonomy). It is organized around a task taxonomy with 55 sub fields grouped into 13 industry clusters covering 1K+ tasks. Current results show that the hardest tier remains far from saturated: across mainstream harness and backbone configurations, the average full pass rate is below 1%. ALE is designed as a living benchmark: its task pool grows continuously as new workflows and industries are onboarded. More broadly, ALE is intended not merely as another leaderboard, but as an instrument for closing the gap between benchmark success and GDP relevant impact. △ Less

Submitted 11 June, 2026; v1 submitted 3 June, 2026; originally announced June 2026.

Comments: Project website: https://agents-last-exam.org Code: https://github.com/rdi-berkeley/agents-last-exam

Quantifying the biophysical properties of stomatocytes in health and disease

Authors: Zhaojie Chai, Jianlu Zheng, He Li, Ming Dao, George Em Karniadakis

Abstract: Hereditary stomatocytosis (HS) comprises red blood cell (RBC) disorders characterized by cup-shaped erythrocytes that respond oppositely to splenectomy: curative in overhydrated HS (OHS) but potentially thrombogenic in dehydrated HS (DHS/xerocytosis). This paradox persists because RBC biomechanics is governed by partly independent parameters--shear modulus, bending rigidity, surface-to-volume rati… ▽ More Hereditary stomatocytosis (HS) comprises red blood cell (RBC) disorders characterized by cup-shaped erythrocytes that respond oppositely to splenectomy: curative in overhydrated HS (OHS) but potentially thrombogenic in dehydrated HS (DHS/xerocytosis). This paradox persists because RBC biomechanics is governed by partly independent parameters--shear modulus, bending rigidity, surface-to-volume ratio (S/V), and cytoplasmic viscosity--that existing assays capture only piecemeal. Here we combine dissipative particle dynamics (DPD) simulations with microfluidic imaging to construct a control discocyte and three stomatocyte models (ST-RBC1-3) at fixed membrane area and decreasing volume (109.7, 101.5, 89.8 fL), spanning the OHS-to-DHS range. Tracing this parameter set through five mechanically orthogonal assays, we find that interendothelial-slit (IES) traversal is geometry-dominated: overhydrated ST-RBC1 requires an order of magnitude higher critical pressure than healthy RBCs, whereas dehydrated ST-RBC3 passes freely. ST-RBC3 nonetheless suppresses membrane tank-treading and raises low-shear whole-blood viscosity by ~29% at physiological haematocrit, comparable to Gaucher-disease hyperviscosity. A funnel-obstacle chip amplifies these differences into a label-free centerline-offset signal predicted to separate all four RBC types (~4.5 standard deviations between extreme phenotypes). These results unite single-cell mechanics, splenic filtration, and hemorheology in one framework, resolve the splenectomy paradox, and point toward microfluidic pre-operative risk stratification in HS. △ Less

Submitted 2 June, 2026; originally announced June 2026.

Comments: 26 pages, 9 figures

MSC Class: 92C05; 76Z05

Spectral Audit of In-Context Operator Networks

Authors: Zhiwei Gao, Liu Yang, George Em Karniadakis

Abstract: Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit fo… ▽ More Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral audit for in-context operator learning. For a fixed prompt, we differentiate the network output with respect to the query function and view the resulting Jacobian as a learned tangent operator. Projecting it onto Fourier modes, we obtain a local spectral characterization of the inferred operator, including frequency-dependent gains, phase structure, and cross-mode coupling. The audit complements standard prediction metrics by testing whether the model reproduces local mechanisms of the underlying PDE operator rather than only outputs. Across benchmarks, the audit reveals distinct operator-level phenomena, including phase transport, viscosity-dependent damping, nonlinear mode coupling, and reaction--diffusion stability structure. It also detects failures partially hidden by prediction-error metrics, including high-frequency degradation, incorrect phase recovery, and prompt--operator inconsistencies. Corrupted or internally inconsistent prompts lead to degraded tangent-operator structure even when pointwise predictions remain partially accurate. Our results suggest that prediction accuracy and local operator fidelity are distinct properties of learned neural operators. Our framework also provides a diagnostic for stability, sensitivity, and operator consistency. △ Less

Submitted 1 June, 2026; originally announced June 2026.

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Authors: Qian Zhang, George Em Karniadakis

Abstract: Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning tas… ▽ More Reconstructing flow fields from sparse measurements is a fundamental problem in fluid mechanics with broad implications for modeling, control, and design. In this work, we propose a novel operator learning framework that leverages the architecture of language models to perform flow reconstruction in a mesh-free manner. We reformulate flow field reconstruction as a sequence-to-sequence learning task, where sparse measurements are treated as context and unobserved locations as queries. Our model learns to reconstruct the full flow field from sparse inputs, effectively capturing spatial correlations and long-range dependencies. We evaluate the proposed approach on four benchmark datasets: (1) two-dimensional vortex street simulations, (2) daily average temperature data across the contiguous United States, (3) three-dimensional blood flow simulations based on dissipative particle dynamics, and (4) three-dimensional turbulent jet flow measurements obtained via particle tracking velocimetry. Across all cases, our method demonstrates competitive reconstruction accuracy, even with highly incomplete data (less than 10\% observed), and achieves efficient performance. The results highlight the potential of language models as robust and scalable tools for scientific data reconstruction, and suggest a promising direction toward the development of foundation models for scientific and engineering applications. △ Less

Submitted 22 May, 2026; originally announced May 2026.

UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning

Authors: Hanli Qiao, George Em Karniadakis, Muhammad Muniruzzaman

Abstract: Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditi… ▽ More Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning. △ Less

Submitted 12 May, 2026; originally announced May 2026.

GRAFT-ATHENA: Self-Improving Agentic Teams for Autonomous Discovery and Evolutionary Numerical Algorithms

Authors: Juan Diego Toscano, Zhaojie Chai, George Em Karniadakis

Abstract: Scientific discovery can be modeled as a sequence of probabilistic decisions that map physical problems to numerical solutions. Recent agentic AI systems automate individual scientific tasks by orchestrating LLM-driven planners, solvers, and evaluators. Each method is a combination of methodological actions, with many viable combinations for any given problem and structural dependencies between ch… ▽ More Scientific discovery can be modeled as a sequence of probabilistic decisions that map physical problems to numerical solutions. Recent agentic AI systems automate individual scientific tasks by orchestrating LLM-driven planners, solvers, and evaluators. Each method is a combination of methodological actions, with many viable combinations for any given problem and structural dependencies between choices. However, existing frameworks treat each problem in isolation, with no shared substrate to accumulate methodological experience across domains. Here we show that GRAFT-ATHENA, a self-improving agentic framework, learns from past problems and autonomously expands its own action space across diverse domains. GRAFT (Graph Reduction to Adaptive Factored Trees) projects combinatorial decision spaces into factored probabilistic trees in which each method is a single path, taking the parameter footprint from exponential to linear. In the lineage of classical Bayesian networks, the factorization is an $I$-map of the policy, and the resulting paths embed as unique fingerprints in a metric space whose closeness lets each new problem learn from similar past ones. On canonical physics-informed machine learning (PIML) benchmarks, GRAFT-ATHENA improves over human and prior agentic baselines, and on production solvers, it tackles complex engineering problems such as reconstructing Mach-10 flow over the Apollo Command Module from a 1968 report and recovering shear-thinning blood-cell rheology. Notably, the system grows its own knowledge substrate, autonomously proposing regularization constraints for ill-posed inverse problems and discovering new numerical methods such as a spectral PINN with exponential convergence. These results provide a foundation for autonomous laboratories that grow more capable with every problem they solve. △ Less

Submitted 11 May, 2026; originally announced May 2026.

NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces

Authors: Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak, George Em Karniadakis

Abstract: The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural ope… ▽ More The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichlet/Neumann boundary conditions, body forces and material properties, among others. We have previously introduced hybridization of classical linear iterative solvers with neural operators for specific geometries, but they tend to not perform well on geometries not previously seen during training. We partially addressed this challenge by introducing the deep operator network Geo-DeepONet and hybridizing it with Krylov-based iterative linear solvers, which, despite learning effectively across arbitrary unstructured meshes without requiring retraining, led to only modest reductions in iterations compared to state-of-the-art preconditioners. In this study we introduce Neural Subspace Proper Orthogonal Decomposition (NSPOD), a multigrid-like deep operator network-based preconditioner which can dramatically reduce the number of iterations needed for convergence in Krylov-based linear iterative solvers, even when compared to state-of-the-art methods such as algebraic multigrid preconditioners. We demonstrate its efficiency via numerical experiments on a linearized version of solid mechanics PDEs applied to unstructured domains obtained from complex CAD geometries. We expect that the findings in this study lead to more efficient hybrid preconditioners that can match, or possibly even surpass, the convergence properties of the current gold standard preconditioning methods for solid mechanics PDEs. △ Less

Submitted 11 May, 2026; v1 submitted 8 May, 2026; originally announced May 2026.

Comments: 17 pages, 9 figures, 3 tables

Multi-fidelity surrogates for mechanics of composites: from co-kriging to multi-fidelity neural networks

Authors: Haizhou Wen, Elham Kiyani, Gang Li, Srikanth Pilla, George Em Karniadakis, Zhen Li

Abstract: Composite materials exhibit strongly hierarchical and anisotropic properties governed by coupled mechanisms spanning constituents, plies, laminates, structures, and manufacturing history. This intrinsic complexity makes predictive modeling of composites expensive, because repeated experiments and high-fidelity simulations are needed to cover large design spaces of material, structure, and manufact… ▽ More Composite materials exhibit strongly hierarchical and anisotropic properties governed by coupled mechanisms spanning constituents, plies, laminates, structures, and manufacturing history. This intrinsic complexity makes predictive modeling of composites expensive, because repeated experiments and high-fidelity simulations are needed to cover large design spaces of material, structure, and manufacturing. Multi-fidelity surrogate modeling addresses this challenge by combining abundant, less expensive data with limited high-accuracy data to recover reliable high-fidelity predictions. This review presents a structured overview of multi-fidelity modeling for composite mechanics, covering Gaussian-process or Kriging-based methods, including co-Kriging, coregionalization models, autoregressive formulations, nonlinear autoregressive Gaussian processes, multi-fidelity deep Gaussian processes, and multi-fidelity neural networks. Their distinctions are examined in terms of cross-fidelity correlation, discrepancy representation, uncertainty quantification, and scalability. Selected examples of their applications to composites are introduced according to the roles that multi-fidelity surrogates play in engineering problems, including forward prediction for rapid exploration of material design spaces, inverse optimization for composite parameter identification and design search under limited high-fidelity access, and workflow integration, where heterogeneous data sources, constraints, and validation requirements determine model utility. Open question discussions highlight recurring challenges specific to composites, such as regime-dependent fidelity gaps associated with nonlinear damage and manufacturing history, mismatches between simulations and experiments, and uncertainty propagation across multi-fidelity models. △ Less

Submitted 4 May, 2026; originally announced May 2026.

Comments: 64 pages, 18 figures. Submitted to Composites Part B: Engineering

Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles

Authors: Khemraj Shukla, Zongren Zou, Theo Kaeufer, Michael Triantafyllou, George Em Karniadakis

Abstract: Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems… ▽ More Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems such as data-driven Reynolds-averaged Navier-Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi-component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE-constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function-space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade-offs between accuracy, computational cost, and uncertainty calibration in physics-informed learning, and offer practical guidance for uncertainty quantification in data-driven turbulence modeling. △ Less

Submitted 18 April, 2026; originally announced April 2026.

Agentic Risk-Aware Set-Based Engineering Design

Authors: Varun Kumar, George Em Karniadakis

Abstract: This paper introduces a multi-agent framework guided by Large Language Models (LLMs) to assist in the early stages of engineering design, a phase often characterized by vast parameter spaces and inherent uncertainty. Operating under a human-in-the-loop paradigm and demonstrated on the canonical problem of aerodynamic airfoil design, the framework employs a team of specialized agents: a Coding Assi… ▽ More This paper introduces a multi-agent framework guided by Large Language Models (LLMs) to assist in the early stages of engineering design, a phase often characterized by vast parameter spaces and inherent uncertainty. Operating under a human-in-the-loop paradigm and demonstrated on the canonical problem of aerodynamic airfoil design, the framework employs a team of specialized agents: a Coding Assistant, a Design Agent, a Systems Engineering Agent, and an Analyst Agent - all coordinated by a human Manager. Integrated within a set-based design philosophy, the process begins with a collaborative phase where the Manager and Coding Assistant develop a suite of validated tools, after which the agents execute a structured workflow to systematically explore and prune a large set of initial design candidates. A key contribution of this work is the explicit integration of formal risk management, employing the Conditional Value-at-Risk (CVaR) as a quantitative metric to filter designs that exhibit a high probability of failing to meet performance requirements, specifically the target coefficient of lift. The framework automates labor-intensive initial exploration through a global sensitivity analysis conducted by the Analyst agent, which generates actionable heuristics to guide the other agents. The process culminates by presenting the human Manager with a curated final set of promising design candidates, augmented with high-fidelity Computational Fluid Dynamics (CFD) simulations. This approach effectively leverages AI to handle high-volume analytical tasks, thereby enhancing the decision-making capability of the human expert in selecting the final, risk-assessed design. △ Less

Submitted 17 April, 2026; originally announced April 2026.

Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks

Authors: Anas Jnini, Elham Kiyani, Khemraj Shukla, Jorge F. Urban, Nazanin Ahmadi Daryakenari, Johannes Muller, Marius Zeinhofer, George Em Karniadakis

Abstract: Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDE… ▽ More Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations. In this work, we present advanced optimization strategies to accelerate the convergence of physics-informed neural networks (PINNs) for challenging partial (PDEs) and ordinary differential equations (ODEs). Specifically, we provide efficient implementations of the Natural Gradient (NG) optimizer, Self-Scaling BFGS and Broyden optimizers, and demonstrate their performance on problems including the Helmholtz equation, Stokes flow, inviscid Burgers equation, Euler equations for high-speed flows, and stiff ODEs arising in pharmacokinetics and pharmacodynamics. Beyond optimizer development, we also propose new PINN-based methods for solving the inviscid Burgers and Euler equations, and compare the resulting solutions against high-order numerical methods to provide a rigorous and fair assessment. Finally, we address the challenge of scaling these quasi-Newton optimizers for batched training, enabling efficient and scalable solutions for large data-driven problems. △ Less

Submitted 6 April, 2026; originally announced April 2026.

Comments: 54 pages, 24 figures

PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods

Authors: Zhen Zhang, Shanqing Liu, Alessandro Alla, Jerome Darbon, George Em Karniadakis

Abstract: We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-o… ▽ More We study physics-informed neural networks (PINNs) as numerical tools for the optimal control of semilinear partial differential equations. We first recall the classical direct and indirect viewpoints for optimal control of PDEs, and then present two PINN formulations: a direct formulation based on minimizing the objective under the state constraint, and an indirect formulation based on the first-order optimality system. For a class of semilinear parabolic equations, we derive the state equation, the adjoint equation, and the stationarity condition in a form consistent with continuous-time Pontryagin-type optimality conditions. We then specialize the framework to an Allen-Cahn control problem and compare three numerical approaches: (i) a discretize-then-optimize adjoint method, (ii) a direct PINN, and (iii) an indirect PINN. Numerical results show that the PINN parameterization has an implicit regularizing effect, in the sense that it tends to produce smoother control profiles. They also indicate that the indirect PINN more faithfully preserves the PDE contraint and optimality structure and yields a more accurate neural approximation than the direct PINN. △ Less

Submitted 6 April, 2026; originally announced April 2026.

Comments: 8 pages, 3 figures

Spectral bias in physics-informed and operator learning: Analysis and mitigation guidelines

Authors: Siavash Khodakarami, Vivek Oommen, Nazanin Ahmadi Daryakenari, Maxim Beekenkamp, George Em Karniadakis

Abstract: Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low-frequency components of the solution are learned significantly faster than high-frequency modes. While spectral bias is often treated a… ▽ More Solving partial differential equations (PDEs) by neural networks as well as Kolmogorov-Arnold Networks (KANs), including physics-informed neural networks (PINNs), physics-informed KANs (PIKANs), and neural operators, are known to exhibit spectral bias, whereby low-frequency components of the solution are learned significantly faster than high-frequency modes. While spectral bias is often treated as an intrinsic representational limitation of neural architectures, its interaction with optimization dynamics and physics-based loss formulations remains poorly understood. In this work, we provide a systematic investigation of spectral bias in physics-informed and operator learning frameworks, with emphasis on the coupled roles of network architecture, activation functions, loss design, and optimization strategy. We quantify spectral bias through frequency-resolved error metrics, Barron-norm diagnostics, and higher-order statistical moments, enabling a unified analysis across elliptic, hyperbolic, and dispersive PDEs. Through diverse benchmark problems, including the Korteweg-de Vries, wave and steady-state diffusion-reaction equations, turbulent flow reconstruction, and earthquake dynamics, we demonstrate that spectral bias is not simply representational but fundamentally dynamical. In particular, second-order optimization methods substantially alter the spectral learning order, enabling earlier and more accurate recovery of high-frequency modes for all PDE types. For neural operators, we further show that spectral bias is dependent on the neural operator architecture and can also be effectively mitigated through spectral-aware loss formulations without increasing the inference cost. △ Less

Submitted 22 February, 2026; originally announced February 2026.

Physics-Informed Laplace Neural Operator for Solving Partial Differential Equations

Authors: Heechang Kim, Qianying Cao, Hyomin Shin, Seungchul Lee, George Em Karniadakis, Minseok Choi

Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-In… ▽ More Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs). However, purely data-driven models often require extensive training data and can generalize poorly, especially in small-data regimes and under unseen (out-of-distribution) input functions that are not represented in the training data. To address these limitations, we propose the Physics-Informed Laplace Neural Operator (PILNO), which enhances the Laplace Neural Operator (LNO) by embedding governing physics into training through PDE, boundary condition, and initial condition residuals. To improve expressivity, we first introduce an Advanced LNO (ALNO) backbone that retains a pole-residue transient representation while replacing the steady-state branch with an FNO-style Fourier multiplier. To make physics-informed training both data-efficient and robust, PILNO further leverages (i) virtual inputs: an unlabeled ensemble of input functions spanning a broad spectral range that provides abundant physics-only supervision and explicitly targets out-of-distribution (OOD) regimes; and (ii) temporal-causality weighting: a time-decaying reweighting of the physics residual that prioritizes early-time dynamics and stabilizes optimization for time-dependent PDEs. Across four representative benchmarks -- Burgers' equation, Darcy flow, a reaction-diffusion system, and a forced KdV equation -- PILNO consistently improves accuracy in small-data settings (e.g., N_train <= 27), reduces run-to-run variability across random seeds, and achieves stronger OOD generalization with respect to input function statistics than purely data-driven baselines. △ Less

Submitted 13 August, 2026; v1 submitted 13 February, 2026; originally announced February 2026.

Comments: 46 pages, 22 figures, 9 tables

Journal ref: Journal of Computational Physics 566 (2026) 115291

An AI-enabled tool for quantifying overlapping red blood cell sickling dynamics in microfluidic assays

Authors: Nikhil Kadivar, Guansheng Li, Jianlu Zheng, Ming Dao, George Em Karniadakis, Mengjia Xu

Abstract: Understanding sickle cell dynamics requires accurate identification of morphological transitions under diverse biophysical conditions, particularly in densely packed and overlapping cell populations. Here, we present an automated deep learning framework that integrates AI-assisted annotation, segmentation, classification, and instance counting to quantify red blood cell (RBC) populations across va… ▽ More Understanding sickle cell dynamics requires accurate identification of morphological transitions under diverse biophysical conditions, particularly in densely packed and overlapping cell populations. Here, we present an automated deep learning framework that integrates AI-assisted annotation, segmentation, classification, and instance counting to quantify red blood cell (RBC) populations across varying density regimes in time-lapse microscopy data. Experimental images were annotated using the Roboflow platform to generate labeled dataset for training an nnU-Net segmentation model. The trained network enables prediction of the temporal evolution of the sickle cell fraction, while a watershed algorithm resolves overlapping cells to enhance quantification accuracy. Despite requiring only a limited amount of labeled data for training, the framework achieves high segmentation performance, effectively addressing challenges associated with scarce manual annotations and cell overlap. By quantitatively tracking dynamic changes in RBC morphology, this approach can more than double the experimental throughput via densely packed cell suspensions, capture drug-dependent sickling behavior, and reveal distinct mechanobiological signatures of cellular morphological evolution. Overall, this AI-driven framework establishes a scalable and reproducible computational platform for investigating cellular biomechanics and assessing therapeutic efficacy in microphysiological systems. △ Less

Submitted 4 February, 2026; v1 submitted 25 January, 2026; originally announced January 2026.

Deep Neural networks for solving high-dimensional parabolic partial differential equations

Authors: Wenzhong Zhang, Zheyuan Hu, Wei Cai, George EM Karniadakis

Abstract: The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh free alternative, enabling the approximation of PDE solutions in tens to thousands of dimensions. This review… ▽ More The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh free alternative, enabling the approximation of PDE solutions in tens to thousands of dimensions. This review provides a tutorial--oriented introduction to neural--network--based methods for solving high dimensional parabolic PDEs, emphasizing conceptual clarity and methodological connections. We organize the literature around three unifying paradigms: (i) PDE residual--based approaches, including physicsinformed neural networks and their high dimensional variants; (ii) stochastic methods derived from Feynman--Kac and backward stochastic differential equation formulations; and (iii) hybrid derivative--free random difference approaches designed to alleviate the computational cost of derivatives in high dimensions. For each paradigm, we outline the underlying mathematical formulation, algorithmic implementation, and practical strengths and limitations. Representative benchmark problems--including Hamilton--Jacobi--Bellman and Black--Scholes equations in up to 1000 dimensions --illustrate the scalability, effectiveness, and accuracy of the methods. The paper concludes with a discussion of open challenges and future directions for reliable and scalable solvers of high dimensional PDEs. △ Less

Submitted 23 January, 2026; v1 submitted 19 January, 2026; originally announced January 2026.

GIMLET: Generalizable and Interpretable Model Learning through Embedded Thermodynamics

Authors: Suguru Shiratori, Elham Kiyani, Khemraj Shukla, George Em Karniadakis

Abstract: We develop a data-driven framework for discovering constitutive relations in models of fluid flow and scalar transport. Under the assumption that velocity and/or scalar fields are measured, our approach infers unknown closure terms in the governing equations as neural networks. The target to be discovered is the constitutive relations only, while the temporal derivative, convective transport terms… ▽ More We develop a data-driven framework for discovering constitutive relations in models of fluid flow and scalar transport. Under the assumption that velocity and/or scalar fields are measured, our approach infers unknown closure terms in the governing equations as neural networks. The target to be discovered is the constitutive relations only, while the temporal derivative, convective transport terms, and pressure-gradient term in the governing equations are prescribed. The formulation is rooted in a variational principle from non-equilibrium thermodynamics, where the dynamics is defined by a free-energy functional and a dissipation functional. The unknown constitutive terms arise as functional derivatives of these functionals with respect to the state variables. To enable a flexible and structured model discovery, the free-energy and dissipation functionals are parameterized using neural networks, while their functional derivatives are obtained via automatic differentiation. This construction enforces thermodynamic consistency by design, guaranteeing monotonic decay of the total free energy and non-negative entropy production. The resulting method, termed GIMLET (Generalizable and Interpretable Model Learning through Embedded Thermodynamics), avoids reliance on a predefined library of candidate functions, unlike sparse regression or symbolic identification approaches. The learned models are generalizable in that functionals identified from one dataset can be transferred to distinct datasets governed by the same underlying equations. Moreover, the inferred free-energy and dissipation functions provide direct physical interpretability of the learned dynamics. The framework is demonstrated on several benchmark systems, including the viscous Burgers equation, the Kuramoto--Sivashinsky equation, and the incompressible Navier--Stokes equations for both Newtonian and non-Newtonian fluids. △ Less

Submitted 30 December, 2025; v1 submitted 22 December, 2025; originally announced December 2025.

Hybrid Iterative Solvers with Geometry-Aware Neural Preconditioners for Parametric PDEs

Authors: Youngkyu Lee, Francesc Levrero Florencio, Jay Pathak, George Em Karniadakis

Abstract: The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address… ▽ More The convergence behavior of classical iterative solvers for parametric partial differential equations (PDEs) is often highly sensitive to the domain and specific discretization of PDEs. Previously, we introduced hybrid solvers by combining the classical solvers with neural operators for a specific geometry 1, but they tend to under-perform in geometries not encountered during training. To address this challenge, we introduce Geo-DeepONet, a geometry-aware deep operator network that incorporates domain information extracted from finite element discretizations. Geo-DeepONet enables accurate operator learning across arbitrary unstructured meshes without requiring retraining. Building on this, we develop a class of geometry-aware hybrid preconditioned iterative solvers by coupling Geo-DeepONet with traditional methods such as relaxation schemes and Krylov subspace algorithms. Through numerical experiments on parametric PDEs posed over diverse unstructured domains, we demonstrate the enhanced robustness and efficiency of the proposed hybrid solvers for multiple real-world applications. △ Less

Submitted 16 December, 2025; originally announced December 2025.

Comments: 19 pages, 10 figures, 3 tables

Journal ref: International Journal for Numerical Methods in Engineering 127 no. 15 (2026) e70393

Kinetic-Mamba: Mamba-Assisted Predictions of Stiff Chemical Kinetics

Authors: Additi Pandey, Liang Wei, Hessam Babaee, George Em Karniadakis

Abstract: Accurate chemical kinetics modeling is essential for combustion simulations, as it governs the evolution of complex reaction pathways and thermochemical states. In this work, we introduce Kinetic-Mamba, a Mamba-based neural operator framework that integrates the expressive power of neural operators with the efficient temporal modeling capabilities of Mamba architectures. The framework comprises th… ▽ More Accurate chemical kinetics modeling is essential for combustion simulations, as it governs the evolution of complex reaction pathways and thermochemical states. In this work, we introduce Kinetic-Mamba, a Mamba-based neural operator framework that integrates the expressive power of neural operators with the efficient temporal modeling capabilities of Mamba architectures. The framework comprises three complementary models: (i) a standalone Mamba model that predicts the time evolution of thermochemical state variables from given initial conditions; (ii) a constrained Mamba model that enforces mass conservation while learning the state dynamics; and (iii) a regime-informed architecture employing two standalone Mamba models to capture dynamics across temperature-dependent regimes. We additionally develop a latent Kinetic-Mamba variant that evolves dynamics in a reduced latent space and reconstructs the full state on the physical manifold. The accuracy and robustness of Kinetic-Mamba was evaluated using both time-decomposition and recursive-prediction strategies. We further assess the extrapolation capabilities of the model on varied out-of-distribution datasets. Computational experiments on Syngas and GRI-Mech 3.0 reaction mechanisms demonstrate that our framework achieves high fidelity in predicting complex kinetic behavior using only the initial conditions of the state variables. △ Less

Submitted 5 April, 2026; v1 submitted 16 December, 2025; originally announced December 2025.

Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Network

Authors: Elham Kiyani, Amit Makarand Deshpande, Madhura Limaye, Zhiwei Gao, Zongren Zou, Sai Aditya Pradeep, Srikanth Pilla, Gang Li, Zhen Li, George Em Karniadakis

Abstract: Fiber reinforcement and polymer matrix respond differently to manufacturing conditions due to mismatch in coefficient of thermal expansion and matrix shrinkage during curing of thermosets. These heterogeneities generate residual stresses over multiple length scales, whose partial release leads to process-induced deformation (PID), requiring accurate prediction and mitigation via optimized non-isot… ▽ More Fiber reinforcement and polymer matrix respond differently to manufacturing conditions due to mismatch in coefficient of thermal expansion and matrix shrinkage during curing of thermosets. These heterogeneities generate residual stresses over multiple length scales, whose partial release leads to process-induced deformation (PID), requiring accurate prediction and mitigation via optimized non-isothermal cure cycles. This study considers a unidirectional AS4 carbon fiber/amine bi-functional epoxy prepreg and models PID using a two-mechanism framework that accounts for thermal expansion/shrinkage and cure shrinkage. The model is validated against manufacturing trials to identify initial and boundary conditions, then used to generate PID responses for a diverse set of non-isothermal cure cycles (time-temperature profiles). Building on this physics-based foundation, we develop a data-driven surrogate based on Deep Operator Networks (DeepONets). A DeepONet is trained on a dataset combining high-fidelity simulations with targeted experimental measurements of PID. We extend this to a Feature-wise Linear Modulation (FiLM) DeepONet, where branch-network features are modulated by external parameters, including the initial degree of cure, enabling prediction of time histories of degree of cure, viscosity, and deformation. Because experimental data are available only at limited time instances (for example, final deformation), we use transfer learning: simulation-trained trunk and branch networks are fixed and only the final layer is updated using measured final deformation. Finally, we augment the framework with Ensemble Kalman Inversion (EKI) to quantify uncertainty under experimental conditions and to support optimization of cure schedules for reduced PID in composites. △ Less

Submitted 1 May, 2026; v1 submitted 14 December, 2025; originally announced December 2025.

Comments: 21 pages, 13 figures

ATHENA: Agentic Team for Hierarchical Evolutionary Numerical Algorithms

Authors: Juan Diego Toscano, Daniel T. Chen, George Em Karniadakis

Abstract: Progress in computational science depends on complex numerical workflows that must faithfully encode physical laws, yet translating conceptual insight into reliable code remains a major bottleneck. Although large language models can generate isolated code fragments, they lack the structured reasoning required to design, verify, and iteratively refine complete scientific pipelines. Here we introduc… ▽ More Progress in computational science depends on complex numerical workflows that must faithfully encode physical laws, yet translating conceptual insight into reliable code remains a major bottleneck. Although large language models can generate isolated code fragments, they lack the structured reasoning required to design, verify, and iteratively refine complete scientific pipelines. Here we introduce ATHENA, an agentic framework explicitly designed to emulate scientific research modeled as a knowledge-driven contextual bandit process. Its core loop separates conceptual policy from numerical realization through expert-derived conceptual scaffolding, enabling principled diagnosis, reformulation, and repair of computational strategies. Across scientific computing and scientific machine learning tasks, ATHENA autonomously derives and correctly applies exact analytical solutions, constructs stable numerical solvers, diagnoses ill-posed formulations, and orchestrates hybrid symbolic-numeric workflows. Quantitatively, ATHENA matches and frequently surpasses the accuracy of expert-authored reference solutions reported in the literature on canonical benchmarks. By reframing computation as an object of agentic reasoning, our framework enables autonomous orchestration of heterogeneous algorithms across scientific domains. △ Less

Submitted 22 June, 2026; v1 submitted 3 December, 2025; originally announced December 2025.

Retrofitting Earth System Models with Cadence-Limited Neural Operator Updates

Authors: Aniruddha Bora, Shixuan Zhang, Khemraj Shukla, Bryce Harrop, George Em. Karniadakis, L. Ruby Leung

Abstract: Coarse resolution, imperfect parameterizations, and uncertain initial states and forcings limit Earth-system model (ESM) predictions. Traditional bias correction via data assimilation improves constrained simulations but offers limited benefit once models run freely. We introduce an operator-learning framework that maps instantaneous model states to bias-correction tendencies and applies them onli… ▽ More Coarse resolution, imperfect parameterizations, and uncertain initial states and forcings limit Earth-system model (ESM) predictions. Traditional bias correction via data assimilation improves constrained simulations but offers limited benefit once models run freely. We introduce an operator-learning framework that maps instantaneous model states to bias-correction tendencies and applies them online during integration. Building on a U-Net backbone, we develop two operator architectures Inception U-Net (IUNet) and a multi-scale network (M\&M) that combine diverse upsampling and receptive fields to capture multiscale nonlinear features under Energy Exascale Earth System Model (E3SM) runtime constraints. Trained on two years E3SM simulations nudged toward ERA5 reanalysis, the operators generalize across height levels and seasons. Both architectures outperform standard U-Net baselines in offline tests, indicating that functional richness rather than parameter count drives performance. In online hybrid E3SM runs, M\&M delivers the most consistent bias reductions across variables and vertical levels. The ML-augmented configurations remain stable and computationally feasible in multi-year simulations, providing a practical pathway for scalable hybrid modeling. Our framework emphasizes long-term stability, portability, and cadence-limited updates, demonstrating the utility of expressive ML operators for learning structured, cross-scale relationships and retrofitting legacy ESMs. △ Less

Submitted 2 December, 2025; originally announced December 2025.

A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers

Authors: Youngkyu Lee, Shanqing Liu, Jerome Darbon, George Em Karniadakis

Abstract: We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional… ▽ More We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities. △ Less

Submitted 11 November, 2025; originally announced November 2025.

Comments: 14 pages, 5 figures, 7 tables

MSC Class: 90C06; 65M55; 65F08; 65F10; 68T07

From LIF to QIF: Toward Differentiable Spiking Neurons for Scientific Machine Learning

Authors: Ruyin Wan, George Em Karniadakis, Panos Stinis

Abstract: Spiking neural networks (SNNs) offer biologically inspired computation but remain underexplored for continuous regression tasks in scientific machine learning. In this work, we introduce and systematically evaluate Quadratic Integrate-and-Fire (QIF) neurons as an alternative to the conventional Leaky Integrate-and-Fire (LIF) model in both directly trained SNNs and ANN-to-SNN conversion frameworks.… ▽ More Spiking neural networks (SNNs) offer biologically inspired computation but remain underexplored for continuous regression tasks in scientific machine learning. In this work, we introduce and systematically evaluate Quadratic Integrate-and-Fire (QIF) neurons as an alternative to the conventional Leaky Integrate-and-Fire (LIF) model in both directly trained SNNs and ANN-to-SNN conversion frameworks. The QIF neuron exhibits smooth and differentiable spiking dynamics, enabling gradient-based training and stable optimization within architectures such as multilayer perceptrons (MLPs), Deep Operator Networks (DeepONets), and Physics-Informed Neural Networks (PINNs). Across benchmarks on function approximation, operator learning, and partial differential equation (PDE) solving, QIF-based networks yield smoother, more accurate, and more stable predictions than their LIF counterparts, which suffer from discontinuous time-step responses and jagged activation surfaces. These results position the QIF neuron as a computational bridge between spiking and continuous-valued deep learning, advancing the integration of neuroscience-inspired dynamics into physics-informed and operator-learning frameworks. △ Less

Submitted 9 November, 2025; originally announced November 2025.

Report number: PNNL-SA-217747

Toward Autonomous Engineering Design: A Knowledge-Guided Multi-Agent Framework

Authors: Varun Kumar, George Em Karniadakis

Abstract: The engineering design process often demands expertise from multiple domains, leading to complex collaborations and iterative refinements. Traditional methods can be resource-intensive and prone to inefficiencies. To address this, we formalize the engineering design process through a multi-agent AI framework that integrates structured design and review loops. The framework introduces specialized k… ▽ More The engineering design process often demands expertise from multiple domains, leading to complex collaborations and iterative refinements. Traditional methods can be resource-intensive and prone to inefficiencies. To address this, we formalize the engineering design process through a multi-agent AI framework that integrates structured design and review loops. The framework introduces specialized knowledge-driven agents that collaborate to generate and refine design candidates. As an exemplar, we demonstrate its application to the aerodynamic optimization of 4-digit NACA airfoils. The framework consists of three key AI agents: a Graph Ontologist, a Design Engineer, and a Systems Engineer. The Graph Ontologist employs a Large Language Model (LLM) to construct two domain-specific knowledge graphs from airfoil design literature. The Systems Engineer, informed by a human manager, formulates technical requirements that guide design generation and evaluation. The Design Engineer leverages the design knowledge graph and computational tools to propose candidate airfoils meeting these requirements. The Systems Engineer reviews and provides feedback both qualitative and quantitative using its own knowledge graph, forming an iterative feedback loop until a design is validated by the manager. The final design is then optimized to maximize performance metrics such as the lift-to-drag ratio. Overall, this work demonstrates how collaborative AI agents equipped with structured knowledge representations can enhance efficiency, consistency, and quality in the engineering design process. △ Less

Submitted 29 December, 2025; v1 submitted 4 November, 2025; originally announced November 2025.

Comments: Added appendices and updated literature review

AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics

Authors: Kamaljyoti Nath, Additi Pandey, Bryan T. Susi, Hessam Babaee, George Em Karniadakis

Abstract: Time integration of stiff systems is a primary source of computational cost in combustion, hypersonics, and other reactive transport systems. This stiffness can introduce time scales significantly smaller than those associated with other physical processes, requiring extremely small time steps in explicit schemes or computationally intensive implicit methods. Consequently, strategies to alleviate… ▽ More Time integration of stiff systems is a primary source of computational cost in combustion, hypersonics, and other reactive transport systems. This stiffness can introduce time scales significantly smaller than those associated with other physical processes, requiring extremely small time steps in explicit schemes or computationally intensive implicit methods. Consequently, strategies to alleviate challenges posed by stiffness are important. While neural operators (DeepONets) can act as surrogates for stiff kinetics, a reliable operator learning strategy is required to appropriately account for differences in error between output variables and samples. Here, we develop AMORE, Adaptive Multi-Output Operator Network, a framework comprising an operator capable of predicting multiple outputs and adaptive loss functions ensuring reliable operator learning. The operator predicts all thermochemical states from given initial conditions. We propose two adaptive loss functions within the framework, considering each state variable's and sample's error to penalize the loss function. We designed the trunk to automatically satisfy Partition of Unity. To enforce unity mass-fraction constraint exactly, we propose an invertible analytical map that transforms the $n$-dimensional species mass-fraction vector into an ($n-1$)-dimensional space. We extend the proposed adaptive loss functions to trunk and branch training in two-step training of DeepONet with multiple outputs. We implemented another unity mass fraction constraint exactly using a softmax function on the predicted mass fraction. We demonstrate efficacy and applicability of our models through two examples: syngas (12 states), GRI-Mech 3.0 (24 active states out of 54). The proposed DeepONet will be a backbone for future CFD studies to accelerate turbulent combustion simulations. AMORE is a general framework, and here, we also demonstrate it for FNO. △ Less

Submitted 15 May, 2026; v1 submitted 14 October, 2025; originally announced October 2025.

Physics-Informed Machine Learning in Biomedical Science and Engineering

Authors: Nazanin Ahmadi, Qianying Cao, Jay D. Humphrey, George Em Karniadakis

Abstract: Physics-informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data-driven methods. Here, we review three main classes of PIML frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growi… ▽ More Physics-informed machine learning (PIML) is emerging as a potentially transformative paradigm for modeling complex biomedical systems by integrating parameterized physical laws with data-driven methods. Here, we review three main classes of PIML frameworks: physics-informed neural networks (PINNs), neural ordinary differential equations (NODEs), and neural operators (NOs), highlighting their growing role in biomedical science and engineering. We begin with PINNs, which embed governing equations into deep learning models and have been successfully applied to biosolid and biofluid mechanics, mechanobiology, and medical imaging among other areas. We then review NODEs, which offer continuous-time modeling, especially suited to dynamic physiological systems, pharmacokinetics, and cell signaling. Finally, we discuss deep NOs as powerful tools for learning mappings between function spaces, enabling efficient simulations across multiscale and spatially heterogeneous biological domains. Throughout, we emphasize applications where physical interpretability, data scarcity, or system complexity make conventional black-box learning insufficient. We conclude by identifying open challenges and future directions for advancing PIML in biomedical science and engineering, including issues of uncertainty quantification, generalization, and integration of PIML and large language models. △ Less

Submitted 6 October, 2025; originally announced October 2025.

Comments: Accepted for publication in the Annual Review of Biomedical Engineering on October 2, 2025

Importance of localized dilatation and distensibility in identifying determinants of thoracic aortic aneurysm with neural operators

Authors: David S. Li, Somdatta Goswami, Qianying Cao, Vivek Oommen, Roland Assi, Jay D. Humphrey, George E. Karniadakis

Abstract: Thoracic aortic aneurysms (TAAs) arise from diverse mechanical and mechanobiological disruptions to the aortic wall that increase the risk of dissection or rupture. Evidence links TAA development to dysfunctions in the aortic mechanotransduction axis, including loss of elastic fiber integrity and cell-matrix connections. Because distinct insults create different mechanical vulnerabilities, there i… ▽ More Thoracic aortic aneurysms (TAAs) arise from diverse mechanical and mechanobiological disruptions to the aortic wall that increase the risk of dissection or rupture. Evidence links TAA development to dysfunctions in the aortic mechanotransduction axis, including loss of elastic fiber integrity and cell-matrix connections. Because distinct insults create different mechanical vulnerabilities, there is a critical need to identify interacting factors that drive progression. Here, we use a finite element framework to generate synthetic TAAs from hundreds of heterogeneous insults spanning varying degrees of elastic fiber damage and impaired mechanosensing. From these simulations, we construct spatial maps of localized dilatation and distensibility to train neural networks that predict the initiating combined insult. We compare several architectures (Deep Operator Networks, UNets, and Laplace Neural Operators) and multiple input data formats to define a standard for future subject-specific modeling. We also quantify predictive performance when networks are trained using only geometric data (dilatation) versus both geometric and mechanical data (dilatation plus distensibility). Across all networks, prediction errors are significantly higher when trained on dilatation alone, underscoring the added value of distensibility information. Among the tested models, UNet consistently provides the highest accuracy across all data formats. These findings highlight the importance of acquiring full-field measurements of both dilatation and distensibility in TAA assessment to reveal the mechanobiological drivers of disease and support the development of personalized treatment strategies. △ Less

Submitted 30 September, 2025; originally announced September 2025.

Process-Informed Forecasting of Complex Thermal Dynamics in Pharmaceutical Manufacturing

Authors: Ramona Rubini, Siavash Khodakarami, Aniruddha Bora, George Em Karniadakis, Michele Dassisti

Abstract: Accurate time-series forecasting for complex physical systems is the backbone of modern industrial monitoring and control, yet deep learning models often lack the physical consistency required in regulated environments.To bridge this gap, we introduce Process-Informed Forecasting (PIF) models for temperature in pharmaceutical lyophilization, embedding deterministic production recipes as macro-stru… ▽ More Accurate time-series forecasting for complex physical systems is the backbone of modern industrial monitoring and control, yet deep learning models often lack the physical consistency required in regulated environments.To bridge this gap, we introduce Process-Informed Forecasting (PIF) models for temperature in pharmaceutical lyophilization, embedding deterministic production recipes as macro-structural priors. We investigate classical methods (e.g., Autoregressive Integrated Moving Average (ARIMA) model) and modern deep learning architectures, including Kolmogorov-Arnold Networks (KANs). We compare three different loss function formulations that integrate a process-informed trajectory prior: a fixed-weight loss, a dynamic uncertainty-based loss, and a Residual-Based Attention (RBA) mechanism. We evaluate all models not only for accuracy and physical consistency but also for robustness to sensor noise. Furthermore, we test the practical generalizability of the best model in a transfer-learning scenario to a new process. Our results show that PIF models outperform their data-driven counterparts in terms of accuracy, physical plausibility and noise resilience, offering a scalable framework for reliable and generalizable forecasting solutions in critical manufacturing. △ Less

Submitted 15 May, 2026; v1 submitted 24 September, 2025; originally announced September 2025.

Journal ref: Engineering Applications of Artificial Intelligence 181 (2026) 115575

A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning

Authors: Juan Diego Toscano, Daniel T. Chen, Vivek Oommen, Jérôme Darbon, George Em Karniadakis

Abstract: Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the residual. Different transformations correspond to distinct objective functionals: exponential weights target the minimization of uniform error, while linear weights re… ▽ More Residual-based adaptive strategies are widely used in scientific machine learning but remain largely heuristic. We introduce a unifying variational framework that formalizes these methods by integrating convex transformations of the residual. Different transformations correspond to distinct objective functionals: exponential weights target the minimization of uniform error, while linear weights recover the minimization of quadratic error. Within this perspective, adaptive weighting is equivalent to selecting sampling distributions that optimize the primal objective, thereby linking discretization choices directly to error metrics. This principled approach yields three benefits: (1) it enables systematic design of adaptive schemes across norms, (2) reduces discretization error through variance reduction of the loss estimator, and (3) enhances learning dynamics by improving the gradient signal-to-noise ratio. Extending the framework to operator learning, we demonstrate substantial performance gains across optimizers and architectures. Our results provide a theoretical justification of residual-based adaptivity and establish a foundation for principled discretization and training strategies. △ Less

Submitted 25 September, 2025; v1 submitted 17 September, 2025; originally announced September 2025.

Learning Nonlinear Responses in PET Bottle Buckling with a Hybrid DeepONet-Transolver Framework

Authors: Varun Kumar, Jing Bi, Cyril Ngo Ngoc, Victor Oancea, George Em Karniadakis

Abstract: Neural surrogates and operator networks for solving partial differential equation (PDE) problems have attracted significant research interest in recent years. However, most existing approaches are limited in their ability to generalize solutions across varying non-parametric geometric domains. In this work, we address this challenge in the context of Polyethylene Terephthalate (PET) bottle bucklin… ▽ More Neural surrogates and operator networks for solving partial differential equation (PDE) problems have attracted significant research interest in recent years. However, most existing approaches are limited in their ability to generalize solutions across varying non-parametric geometric domains. In this work, we address this challenge in the context of Polyethylene Terephthalate (PET) bottle buckling analysis, a representative packaging design problem conventionally solved using computationally expensive finite element analysis (FEA). We introduce a hybrid DeepONet-Transolver framework that simultaneously predicts nodal displacement fields and the time evolution of reaction forces during top load compression. Our methodology is evaluated on two families of bottle geometries parameterized by two and four design variables. Training data is generated using nonlinear FEA simulations in Abaqus for 254 unique designs per family. The proposed framework achieves mean relative $L^{2}$ errors of 2.5-13% for displacement fields and approximately 2.4% for time-dependent reaction forces for the four-parameter bottle family. Point-wise error analyses further show absolute displacement errors on the order of $10^{-4}$-$10^{-3}$, with the largest discrepancies confined to localized geometric regions. Importantly, the model accurately captures key physical phenomena, such as buckling behavior, across diverse bottle geometries. These results highlight the potential of our framework as a scalable and computationally efficient surrogate, particularly for multi-task predictions in computational mechanics and applications requiring rapid design evaluation. △ Less

Submitted 16 September, 2025; originally announced September 2025.

Learning Turbulent Flows with Generative Models: Super-resolution, Forecasting, and Sparse Flow Reconstruction

Authors: Vivek Oommen, Siavash Khodakarami, Aniruddha Bora, Zhicheng Wang, George Em Karniadakis

Abstract: Neural operators are promising surrogates for dynamical systems but when trained with standard L2 losses they tend to oversmooth fine-scale turbulent structures. Here, we show that combining operator learning with generative modeling overcomes this limitation. We consider three practical turbulent-flow challenges where conventional neural operators fail: spatio-temporal super-resolution, forecasti… ▽ More Neural operators are promising surrogates for dynamical systems but when trained with standard L2 losses they tend to oversmooth fine-scale turbulent structures. Here, we show that combining operator learning with generative modeling overcomes this limitation. We consider three practical turbulent-flow challenges where conventional neural operators fail: spatio-temporal super-resolution, forecasting, and sparse flow reconstruction. For Schlieren jet super-resolution, an adversarially trained neural operator (adv-NO) reduces the energy-spectrum error by 15x while preserving sharp gradients at neural operator-like inference cost. For 3D homogeneous isotropic turbulence, adv-NO trained on only 160 timesteps from a single trajectory forecasts accurately for five eddy-turnover times and offers 114x wall-clock speed-up at inference than the baseline diffusion-based forecasters, enabling near-real-time rollouts. For reconstructing cylinder wake flows from highly sparse Particle Tracking Velocimetry-like inputs, a conditional generative model infers full 3D velocity and pressure fields with correct phase alignment and statistics. These advances enable accurate reconstruction and forecasting at low compute cost, bringing near-real-time analysis and control within reach in experimental and computational fluid mechanics. See our project page: https://vivekoommen.github.io/Gen4Turb/ △ Less

Submitted 10 September, 2025; originally announced September 2025.

Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method

Authors: Alena Kopaničáková, Youngkyu Lee, George Em Karniadakis

Abstract: We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network~(DeepON… ▽ More We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network~(DeepONet). To this aim, we introduce two complementary approaches for assembling the deflation operators. The first approach approximates near-null space vectors of the discrete PDE operator using the basis functions learned by the DeepONet. The second approach directly leverages solutions predicted by the DeepONet. To further enhance convergence, we also propose several strategies for prescribing the sparsity pattern of the deflation operator. A comprehensive set of numerical experiments encompassing steady-state, time-dependent, scalar, and vector-valued problems posed on both structured and unstructured geometries is presented and demonstrates the effectiveness of the proposed DeepONet-based deflated PCG method, as well as its generalization across a wide range of model parameters and problem resolutions. △ Less

Submitted 31 July, 2025; originally announced August 2025.

Comments: 31 pages

MSC Class: 65M55; 68T05; 49K20

Crack Path Prediction with Operator Learning using Discrete Particle System data Generation

Authors: Elham Kiyani, Venkatesh Ananchaperumal, Ahmad Peyvan, Mahendaran Uchimali, Gang Li, George Em Karniadakis

Abstract: Accurately modeling crack propagation is critical for predicting failure in engineering materials and structures, where small cracks can rapidly evolve and cause catastrophic damage. The interaction of cracks with discontinuities, such as holes, significantly affects crack deflection and arrest. Recent developments in discrete particle systems with multibody interactions based on constitutive beha… ▽ More Accurately modeling crack propagation is critical for predicting failure in engineering materials and structures, where small cracks can rapidly evolve and cause catastrophic damage. The interaction of cracks with discontinuities, such as holes, significantly affects crack deflection and arrest. Recent developments in discrete particle systems with multibody interactions based on constitutive behavior have demonstrated the ability to capture crack nucleation and evolution without relying on continuum assumptions. In this work, we use data from Constitutively Informed Particle Dynamics (CPD) simulations to train operator learning models, specifically Deep Operator Networks (DeepONets), which learn mappings between function spaces instead of finite-dimensional vectors. We explore two DeepONet variants: vanilla and Fusion DeepONet, for predicting time-evolving crack propagation in specimens with varying geometries. Three representative cases are studied: (i) varying notch height without active fracture; and (ii) and (iii) combinations of notch height and hole radius where dynamic fracture occurs on irregular discrete meshes. The models are trained using geometric inputs in the branch network and spatial-temporal coordinates in the trunk network. Results show that Fusion DeepONet consistently outperforms the vanilla variant, with more accurate predictions especially in non-fracturing cases. Fracture-driven scenarios involving displacement and crack evolution remain more challenging. These findings highlight the potential of Fusion DeepONet to generalize across complex, geometry-varying, and time-dependent crack propagation phenomena. △ Less

Submitted 10 September, 2025; v1 submitted 15 May, 2025; originally announced June 2025.

Comments: 22 pages, 14 figures

FMEnets: Flow, Material, and Energy networks for non-ideal plug flow reactor design

Authors: Chenxi Wu, Juan Diego Toscano, Khemraj Shukla, Yingjie Chen, Ali Shahmohammadi, Edward Raymond, Thomas Toupy, Neda Nazemifard, Charles Papageorgiou, George Em Karniadakis

Abstract: We propose FMEnets, a physics-informed machine learning framework for the design and analysis of non-ideal plug flow reactors. FMEnets integrates the fundamental governing equations (Navier-Stokes for fluid flow, material balance for reactive species transport, and energy balance for temperature distribution) into a unified multi-scale network model. The framework is composed of three interconnect… ▽ More We propose FMEnets, a physics-informed machine learning framework for the design and analysis of non-ideal plug flow reactors. FMEnets integrates the fundamental governing equations (Navier-Stokes for fluid flow, material balance for reactive species transport, and energy balance for temperature distribution) into a unified multi-scale network model. The framework is composed of three interconnected sub-networks with independent optimizers that enable both forward and inverse problem-solving. In the forward mode, FMEnets predicts velocity, pressure, species concentrations, and temperature profiles using only inlet and outlet information. In the inverse mode, FMEnets utilizes sparse multi-residence-time measurements to simultaneously infer unknown kinetic parameters and states. FMEnets can be implemented either as FME-PINNs, which employ conventional multilayer perceptrons, or as FME-KANs, based on Kolmogorov-Arnold Networks. Comprehensive ablation studies highlight the critical role of the FMEnets architecture in achieving accurate predictions. Specifically, FME-KANs are more robust to noise than FME-PINNs, although both representations are comparable in accuracy and speed in noise-free conditions. The proposed framework is applied to three different sets of reaction scenarios and is compared with finite element simulations. FMEnets effectively captures the complex interactions, achieving relative errors less than 2.5% for the unknown kinetic parameters. The new network framework not only provides a computationally efficient alternative for reactor design and optimization, but also opens new avenues for integrating empirical correlations, limited and noisy experimental data, and fundamental physical equations to guide reactor design. △ Less

Submitted 9 May, 2025; originally announced May 2025.

Equilibrium Conserving Neural Operators for Super-Resolution Learning

Authors: Vivek Oommen, Andreas E. Robertson, Daniel Diaz, Coleman Alleman, Zhen Zhang, Anthony D. Rollett, George E. Karniadakis, Rémi Dingreville

Abstract: Neural surrogate solvers can estimate solutions to partial differential equations in physical problems more efficiently than standard numerical methods, but require extensive high-resolution training data. In this paper, we break this limitation; we introduce a framework for super-resolution learning in solid mechanics problems. Our approach allows one to train a high-resolution neural network usi… ▽ More Neural surrogate solvers can estimate solutions to partial differential equations in physical problems more efficiently than standard numerical methods, but require extensive high-resolution training data. In this paper, we break this limitation; we introduce a framework for super-resolution learning in solid mechanics problems. Our approach allows one to train a high-resolution neural network using only low-resolution data. Our Equilibrium Conserving Operator (ECO) architecture embeds known physics directly into the network to make up for missing high-resolution information during training. We evaluate this ECO-based super-resolution framework that strongly enforces conservation-laws in the predicted solutions on two working examples: embedded pores in a homogenized matrix and randomly textured polycrystalline materials. ECO eliminates the reliance on high-fidelity data and reduces the upfront cost of data collection by two orders of magnitude, offering a robust pathway for resource-efficient surrogate modeling in materials modeling. ECO is readily generalizable to other physics-based problems. △ Less

Submitted 17 April, 2025; originally announced April 2025.

Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology

Authors: Nazanin Ahmadi Daryakenari, Khemraj Shukla, George Em Karniadakis

Abstract: Physics-Informed Kolmogorov-Arnold Networks (PIKANs) are gaining attention as an effective counterpart to the original multilayer perceptron-based Physics-Informed Neural Networks (PINNs). Both representation models can address inverse problems and facilitate gray-box system identification. However, a comprehensive understanding of their performance in terms of accuracy and speed remains underexpl… ▽ More Physics-Informed Kolmogorov-Arnold Networks (PIKANs) are gaining attention as an effective counterpart to the original multilayer perceptron-based Physics-Informed Neural Networks (PINNs). Both representation models can address inverse problems and facilitate gray-box system identification. However, a comprehensive understanding of their performance in terms of accuracy and speed remains underexplored. In particular, we introduce a modified PIKAN architecture, tanh-cPIKAN, which is based on Chebyshev polynomials for parametrization of the univariate functions with an extra nonlinearity for enhanced performance. We then present a systematic investigation of how choices of the optimizer, representation, and training configuration influence the performance of PINNs and PIKANs in the context of systems pharmacology modeling. We benchmark a wide range of first-order, second-order, and hybrid optimizers, including various learning rate schedulers. We use the new Optax library to identify the most effective combinations for learning gray-boxes under ill-posed, non-unique, and data-sparse conditions. We examine the influence of model architecture (MLP vs. KAN), numerical precision (single vs. double), the need for warm-up phases for second-order methods, and sensitivity to the initial learning rate. We also assess the optimizer scalability for larger models and analyze the trade-offs introduced by JAX in terms of computational efficiency and numerical accuracy. Using two representative systems pharmacology case studies - a pharmacokinetics model and a chemotherapy drug-response model - we offer practical guidance on selecting optimizers and representation models/architectures for robust and efficient gray-box discovery. Our findings provide actionable insights for improving the training of physics-informed networks in biomedical applications and beyond. △ Less

Submitted 14 November, 2025; v1 submitted 9 April, 2025; originally announced April 2025.

MSC Class: 35R30 (Primary); 65M32; 92C50 (Secondary) ACM Class: I.2.6; G.1.7; G.1.10

Mitigating Spectral Bias in Neural Operators via High-Frequency Scaling for Physical Systems

Authors: Siavash Khodakarami, Vivek Oommen, Aniruddha Bora, George Em Karniadakis

Abstract: Neural operators have emerged as powerful surrogates for modeling complex physical problems. However, they suffer from spectral bias making them oblivious to high-frequency modes, which are present in multiscale physical systems. Therefore, they tend to produce over-smoothed solutions, which is particularly problematic in modeling turbulence and for systems with intricate patterns and sharp gradie… ▽ More Neural operators have emerged as powerful surrogates for modeling complex physical problems. However, they suffer from spectral bias making them oblivious to high-frequency modes, which are present in multiscale physical systems. Therefore, they tend to produce over-smoothed solutions, which is particularly problematic in modeling turbulence and for systems with intricate patterns and sharp gradients such as multi-phase flow systems. In this work, we introduce a new approach named high-frequency scaling (HFS) to mitigate spectral bias in convolutional-based neural operators. By integrating HFS with proper variants of UNet neural operators, we demonstrate a higher prediction accuracy by mitigating spectral bias in single and two-phase flow problems. Unlike Fourier-based techniques, HFS is directly applied to the latent space, thus eliminating the computational cost associated with the Fourier transform. Additionally, we investigate alternative spectral bias mitigation through diffusion models conditioned on neural operators. While the diffusion model integrated with the standard neural operator may still suffer from significant errors, these errors are substantially reduced when the diffusion model is integrated with a HFS-enhanced neural operator. △ Less

Submitted 17 March, 2025; originally announced March 2025.

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

Authors: Zongren Zou, Zhicheng Wang, George Em Karniadakis

Abstract: We explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty is giving appropriate initial conditions or initial gue… ▽ More We explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty is giving appropriate initial conditions or initial guesses, to which the widely used time-marching schemes and Newton's iteration method are very sensitive in finding solutions for complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first reveal that PINNs, when combined with random initialization and deep ensemble method -- originally developed for uncertainty quantification -- can effectively uncover multiple solutions to nonlinear ordinary and partial differential equations (ODEs/PDEs). Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen-Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear differential equations. △ Less

Submitted 8 March, 2025; originally announced March 2025.

DeepSeek vs. ChatGPT vs. Claude: A Comparative Study for Scientific Computing and Scientific Machine Learning Tasks

Authors: Qile Jiang, Zhiwei Gao, George Em Karniadakis

Abstract: Large Language Models (LLMs) have emerged as powerful tools for tackling a wide range of problems, including those in scientific computing, particularly in solving partial differential equations (PDEs). However, different models exhibit distinct strengths and preferences, resulting in varying levels of performance. In this paper, we compare the capabilities of the most advanced LLMs--DeepSeek, Cha… ▽ More Large Language Models (LLMs) have emerged as powerful tools for tackling a wide range of problems, including those in scientific computing, particularly in solving partial differential equations (PDEs). However, different models exhibit distinct strengths and preferences, resulting in varying levels of performance. In this paper, we compare the capabilities of the most advanced LLMs--DeepSeek, ChatGPT, and Claude--along with their reasoning-optimized versions in addressing computational challenges. Specifically, we evaluate their proficiency in solving traditional numerical problems in scientific computing as well as leveraging scientific machine learning techniques for PDE-based problems. We designed all our experiments so that a non-trivial decision is required, e.g. defining the proper space of input functions for neural operator learning. Our findings show that reasoning and hybrid-reasoning models consistently and significantly outperform non-reasoning ones in solving challenging problems, with ChatGPT o3-mini-high generally offering the fastest reasoning speed. △ Less

Submitted 8 March, 2025; v1 submitted 24 February, 2025; originally announced February 2025.

Connecting the geometry and dynamics of many-body complex systems with message passing neural operators

Authors: Nicholas A. Gabriel, Neil F. Johnson, George Em Karniadakis

Abstract: The relationship between scale transformations and dynamics established by renormalization group techniques is a cornerstone of modern physical theories, from fluid mechanics to elementary particle physics. Integrating renormalization group methods into neural operators for many-body complex systems could provide a foundational inductive bias for learning their effective dynamics, while also uncov… ▽ More The relationship between scale transformations and dynamics established by renormalization group techniques is a cornerstone of modern physical theories, from fluid mechanics to elementary particle physics. Integrating renormalization group methods into neural operators for many-body complex systems could provide a foundational inductive bias for learning their effective dynamics, while also uncovering multiscale organization. We introduce a scalable AI framework, ROMA (Renormalized Operators with Multiscale Attention), for learning multiscale evolution operators of many-body complex systems. In particular, we develop a renormalization procedure based on neural analogs of the geometric and laplacian renormalization groups, which can be co-learned with neural operators. An attention mechanism is used to model multiscale interactions by connecting geometric representations of local subgraphs and dynamical operators. We apply this framework in challenging conditions: large systems of more than 1M nodes, long-range interactions, and noisy input-output data for two contrasting examples: Kuramoto oscillators and Burgers-like social dynamics. We demonstrate that the ROMA framework improves scalability and positive transfer between forecasting and effective dynamics tasks compared to state-of-the-art operator learning techniques, while also giving insight into multiscale interactions. Additionally, we investigate power law scaling in the number of model parameters, and demonstrate a departure from typical power law exponents in the presence of hierarchical and multiscale interactions. △ Less

Submitted 21 February, 2025; originally announced February 2025.

Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks

Authors: Elham Kiyani, Khemraj Shukla, Jorge F. Urbán, Jérôme Darbon, George Em Karniadakis

Abstract: Physics-Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network's training process as soft constraints, becoming an important component of the scientific machine learning (SciML) ecosystem. More recently, physics-informed Kolmogorv-Arnold networks (PIKANs) have also shown to be effective and… ▽ More Physics-Informed Neural Networks (PINNs) have revolutionized the computation of PDE solutions by integrating partial differential equations (PDEs) into the neural network's training process as soft constraints, becoming an important component of the scientific machine learning (SciML) ecosystem. More recently, physics-informed Kolmogorv-Arnold networks (PIKANs) have also shown to be effective and comparable in accuracy with PINNs. In their current implementation, both PINNs and PIKANs are mainly optimized using first-order methods like Adam, as well as quasi-Newton methods such as BFGS and its low-memory variant, L-BFGS. However, these optimizers often struggle with highly non-linear and non-convex loss landscapes, leading to challenges such as slow convergence, local minima entrapment, and (non)degenerate saddle points. In this study, we investigate the performance of Self-Scaled BFGS (SSBFGS), Self-Scaled Broyden (SSBroyden) methods and other advanced quasi-Newton schemes, including BFGS and L-BFGS with different line search strategies. These methods dynamically rescale updates based on historical gradient information, thus enhancing training efficiency and accuracy. We systematically compare these optimizers using both PINNs and PIKANs on key challenging PDEs, including the Burgers, Allen-Cahn, Kuramoto-Sivashinsky, Ginzburg-Landau, and Stokes equations. Additionally, we evaluate the performance of SSBFGS and SSBroyden for Deep Operator Network (DeepONet) architectures, demonstrating their effectiveness for data-driven operator learning. Our findings provide state-of-the-art results with orders-of-magnitude accuracy improvements without the use of adaptive weights or any other enhancements typically employed in PINNs. △ Less

Submitted 8 February, 2026; v1 submitted 22 January, 2025; originally announced January 2025.

Comments: 43 pages, 32 figures

Automatic selection of the best neural architecture for time series forecasting

Authors: Qianying Cao, Shanqing Liu, Alan John Varghese, Jerome Darbon, Michael Triantafyllou, George Em Karniadakis

Abstract: Time series forecasting plays a pivotal role in a wide range of applications, including weather prediction, healthcare, structural health monitoring, predictive maintenance, energy systems, and financial markets. While models such as LSTM, GRU, Transformers, and State-Space Models (SSMs) have become standard tools in this domain, selecting the optimal architecture remains a challenge. Performance… ▽ More Time series forecasting plays a pivotal role in a wide range of applications, including weather prediction, healthcare, structural health monitoring, predictive maintenance, energy systems, and financial markets. While models such as LSTM, GRU, Transformers, and State-Space Models (SSMs) have become standard tools in this domain, selecting the optimal architecture remains a challenge. Performance comparisons often depend on evaluation metrics and the datasets under analysis, making the choice of a universally optimal model controversial. In this work, we introduce a flexible automated framework for time series forecasting that systematically designs and evaluates diverse network architectures by integrating LSTM, GRU, multi-head Attention, and SSM blocks. Using a multi-objective optimization approach, our framework determines the number, sequence, and combination of blocks to align with specific requirements and evaluation objectives. From the resulting Pareto-optimal architectures, the best model for a given context is selected via a user-defined preference function. We validate our framework across four distinct real-world applications. Results show that a single-layer GRU or LSTM is usually optimal when minimizing training time alone. However, when maximizing accuracy or balancing multiple objectives, the best architectures are often composite designs incorporating multiple block types in specific configurations. By employing a weighted preference function, users can resolve trade-offs between objectives, revealing novel, context-specific optimal architectures. Our findings underscore that no single neural architecture is universally optimal for time series forecasting. Instead, the best-performing model emerges as a data-driven composite architecture tailored to user-defined criteria and evaluation objectives. △ Less

Submitted 1 April, 2026; v1 submitted 21 January, 2025; originally announced January 2025.

Comments: 35 pages, 8 figures

Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks

Authors: Zhiwei Gao, George Em Karniadakis

Abstract: Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov-Arnold networks (KANs) offer an alternative solution with fewer parameters. Ho… ▽ More Uncertainty quantification (UQ) plays a pivotal role in scientific machine learning, especially when surrogate models are used to approximate complex systems. Although multilayer perceptions (MLPs) are commonly employed as surrogates, they often suffer from overfitting due to their large number of parameters. Kolmogorov-Arnold networks (KANs) offer an alternative solution with fewer parameters. However, gradient-based inference methods, such as Hamiltonian Monte Carlo (HMC), may result in computational inefficiency when applied to KANs, especially for large-scale datasets, due to the high cost of back-propagation. To address these challenges, we propose a novel approach, combining the dropout Tikhonov ensemble Kalman inversion (DTEKI) with Chebyshev KANs. This gradient-free method effectively mitigates overfitting and enhances numerical stability. Additionally, we incorporate the active subspace method to reduce the parameter-space dimensionality, allowing us to improve the accuracy of predictions and obtain more reliable uncertainty estimates. Extensive experiments demonstrate the efficacy of our approach in various test cases, including scenarios with large datasets and high noise levels. Our results show that the new method achieves comparable or better accuracy, much higher efficiency as well as stability compared to HMC, in addition to scalability. Moreover, by leveraging the low-dimensional parameter subspace, our method preserves prediction accuracy while substantially reducing further the computational cost. △ Less

Submitted 20 January, 2025; v1 submitted 14 January, 2025; originally announced January 2025.

Comments: 28 pages, 19 figures

Operator Learning for Reconstructing Flow Fields from Sparse Measurements: an Energy Transformer Approach

Authors: Qian Zhang, Dmitry Krotov, George Em Karniadakis

Abstract: Machine learning methods have shown great success in various scientific areas, including fluid mechanics. However, reconstruction problems, where full velocity fields must be recovered from partial observations, remain challenging. In this paper, we propose a novel operator learning framework for solving reconstruction problems by using the Energy Transformer (ET), an architecture inspired by asso… ▽ More Machine learning methods have shown great success in various scientific areas, including fluid mechanics. However, reconstruction problems, where full velocity fields must be recovered from partial observations, remain challenging. In this paper, we propose a novel operator learning framework for solving reconstruction problems by using the Energy Transformer (ET), an architecture inspired by associative memory models. We formulate reconstruction as a mapping from incomplete observed data to full reconstructed fields. The method is validated on three fluid mechanics examples using diverse types of data: (1) unsteady 2D vortex street in flow past a cylinder using simulation data; (2) high-speed under-expanded impinging supersonic jets impingement using Schlieren imaging; and (3) 3D turbulent jet flow using particle tracking. The results demonstrate the ability of ET to accurately reconstruct complex flow fields from highly incomplete data (90\% missing), even for noisy experimental measurements, with fast training and inference on a single GPU. This work provides a promising new direction for tackling reconstruction problems in fluid mechanics and other areas in mechanics, geophysics, weather prediction, and beyond. △ Less

Submitted 2 January, 2025; originally announced January 2025.

DeepVIVONet: Using deep neural operators to optimize sensor locations with application to vortex-induced vibrations

Authors: Ruyin Wan, Ehsan Kharazmi, Michael S Triantafyllou, George Em Karniadakis

Abstract: We introduce DeepVIVONet, a new framework for optimal dynamic reconstruction and forecasting of the vortex-induced vibrations (VIV) of a marine riser, using field data. We demonstrate the effectiveness of DeepVIVONet in accurately reconstructing the motion of an off--shore marine riser by using sparse spatio-temporal measurements. We also show the generalization of our model in extrapolating to ot… ▽ More We introduce DeepVIVONet, a new framework for optimal dynamic reconstruction and forecasting of the vortex-induced vibrations (VIV) of a marine riser, using field data. We demonstrate the effectiveness of DeepVIVONet in accurately reconstructing the motion of an off--shore marine riser by using sparse spatio-temporal measurements. We also show the generalization of our model in extrapolating to other flow conditions via transfer learning, underscoring its potential to streamline operational efficiency and enhance predictive accuracy. The trained DeepVIVONet serves as a fast and accurate surrogate model for the marine riser, which we use in an outer--loop optimization algorithm to obtain the optimal locations for placing the sensors. Furthermore, we employ an existing sensor placement method based on proper orthogonal decomposition (POD) to compare with our data-driven approach. We find that that while POD offers a good approach for initial sensor placement, DeepVIVONet's adaptive capabilities yield more precise and cost-effective configurations. △ Less

Submitted 7 January, 2025; originally announced January 2025.

Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows

Authors: Ahmad Peyvan, Varun Kumar, George Em Karniadakis

Abstract: Shape optimization is essential in aerospace vehicle design, including reentry systems, and propulsion system components, as it directly influences aerodynamic efficiency, structural integrity, and overall mission success. Rapid and accurate prediction of external and internal flows accelerates design iterations. To this end, we develop a new variant of DeepONet, called Fusion-DeepONet as a fast s… ▽ More Shape optimization is essential in aerospace vehicle design, including reentry systems, and propulsion system components, as it directly influences aerodynamic efficiency, structural integrity, and overall mission success. Rapid and accurate prediction of external and internal flows accelerates design iterations. To this end, we develop a new variant of DeepONet, called Fusion-DeepONet as a fast surrogate model for geometry-dependent hypersonic and supersonic flow fields. We evaluated Fusion-DeepONet in learning two external hypersonic flows and a supersonic shape-dependent internal flow problem. First, we compare the performance of Fusion-DeepONet with state-of-the-art neural operators to learn inviscid hypersonic flow around semi-elliptic blunt bodies for two grid types: uniform Cartesian and irregular grids. Fusion-DeepONet provides comparable accuracy to parameter-conditioned U-Net on uniform grids while outperforming MeshGraphNet and Vanilla-DeepONet on irregular grids. Fusion-DeepONet requires significantly fewer trainable parameters than U-Net, MeshGraphNet, and FNO. For the second hypersonic problem, we set up Fusion-DeepONet to map from geometry and free stream Mach number to the temperature field around a reentry capsule traveling at hypersonic speed. This fast surrogate is then improved to predict the spatial derivative of the temperature, resulting in an accurate prediction of heat flux at the surfaces of the capsule. To enhance the accuracy of spatial derivative prediction, we introduce a derivative-enhanced loss term with the least computation overhead. For the third problem, we show that Fusion-DeepONet outperforms MeshGraphNet in learning geometry-dependent supersonic flow in a converging-diverging nozzle configuration. For all the problems, we used high-fidelity simulations with a high-order entropy-stable DGSEM solver to generate training datasets with limited samples. △ Less

Submitted 23 May, 2025; v1 submitted 3 January, 2025; originally announced January 2025.