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Learning Lattice Parameters from Powder X-Ray Diffraction Data Using Invariants
Authors:
Elyssa Hofgard,
Kyucheol Min,
Nofit Segal,
David W. Mittan-Moreau,
Aria Mansouri Tehrani,
Vanessa Oklejas,
Jigyasa Nigam,
Daniel W. Paley,
Aaron S. Brewster,
Tess Smidt
Abstract:
We present a machine learning (ML) method to determine unit cell parameters from powder X-Ray diffraction (XRD) data using a novel invariant lattice representation. In ML, the data representation used can have a substantial impact on the prediction quality. Previous approaches have directly predicted lattice parameters ($a,b,c,α,β,γ$) from XRD inputs. However, these parameters depend strongly on t…
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We present a machine learning (ML) method to determine unit cell parameters from powder X-Ray diffraction (XRD) data using a novel invariant lattice representation. In ML, the data representation used can have a substantial impact on the prediction quality. Previous approaches have directly predicted lattice parameters ($a,b,c,α,β,γ$) from XRD inputs. However, these parameters depend strongly on the unit cell reduction or convention used. In this work, we construct an invariant representation of the reciprocal lattice that is independent of primitive cell convention, based on the bispectrum--a descriptor built from spherical harmonic projections of lattice points. The calculation of the lattice bispectrum is differentiable, and we demonstrate how to invert it using a dynamic programming approach. We show that when fixing ML model architecture, using the lattice bispectrum as the ML target rather than the unit cell parameters leads to more accurate lattice parameter predictions. For example, using the MP-20 dataset, the bispectrum reduces length mean absolute percentage error (MAPE) from 11.18% to 2.44% and angle MAPE from 12.74% to 3.07% compared to direct prediction with the same model architecture. We additionally benchmark our approach against pre-existing XRD to crystal structure models such as Crystalyze and assess its performance on the experimental RRUFF dataset. Beyond unit cell representation, we anticipate this invariant lattice representation could serve more broadly as a geometry-aware target for other crystallographic machine learning tasks such as structure generation.
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Submitted 23 July, 2026;
originally announced July 2026.
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PFT: Phonon Fine-tuning for Machine Learned Interatomic Potentials
Authors:
Teddy Koker,
Abhijeet Gangan,
Mit Kotak,
Jaime Marian,
Tess Smidt
Abstract:
Many materials properties depend on higher-order derivatives of the potential energy surface, yet machine learned interatomic potentials (MLIPs) trained with a standard loss on energy, force, and stress errors can exhibit error in curvature, degrading the prediction of vibrational properties. We introduce phonon fine-tuning (PFT), which directly supervises second-order force constants of materials…
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Many materials properties depend on higher-order derivatives of the potential energy surface, yet machine learned interatomic potentials (MLIPs) trained with a standard loss on energy, force, and stress errors can exhibit error in curvature, degrading the prediction of vibrational properties. We introduce phonon fine-tuning (PFT), which directly supervises second-order force constants of materials by matching MLIP energy Hessians to DFT-computed force constants from finite displacement phonon calculations. To scale to large supercells, PFT stochastically samples Hessian columns and computes the loss with a single Hessian-vector product. We also use a simple co-training scheme to incorporate upstream data to mitigate catastrophic forgetting. On the MDR Phonon benchmark, PFT improves Nequix MP by 55% on average across phonon thermodynamic properties and achieves state-of-the-art accuracy among models trained on Materials Project trajectories. PFT also generalizes to improve properties beyond second-derivatives, improving thermal conductivity predictions that rely on third-order derivatives of the potential energy.
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Submitted 30 May, 2026; v1 submitted 12 January, 2026;
originally announced January 2026.
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Phonon predictions with E(3)-equivariant graph neural networks
Authors:
Shiang Fang,
Mario Geiger,
Joseph G. Checkelsky,
Tess Smidt
Abstract:
We present an equivariant neural network for predicting vibrational and phonon modes of molecules and periodic crystals, respectively. These predictions are made by evaluating the second derivative Hessian matrices of the learned energy model that is trained with the energy and force data. Using this method, we are able to efficiently predict phonon dispersion and the density of states for inorgan…
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We present an equivariant neural network for predicting vibrational and phonon modes of molecules and periodic crystals, respectively. These predictions are made by evaluating the second derivative Hessian matrices of the learned energy model that is trained with the energy and force data. Using this method, we are able to efficiently predict phonon dispersion and the density of states for inorganic crystal materials. For molecules, we also derive the symmetry constraints for IR/Raman active modes by analyzing the phonon mode irreducible representations. Additionally, we demonstrate that using Hessian as a new type of higher-order training data improves energy models beyond models that only use lower-order energy and force data. With this second derivative approach, one can directly relate the energy models to the experimental observations for the vibrational properties. This approach further connects to a broader class of physical observables with a generalized energy model that includes external fields.
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Submitted 17 March, 2024;
originally announced March 2024.
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Quantifying chemical short-range order in metallic alloys
Authors:
Killian Sheriff,
Yifan Cao,
Tess Smidt,
Rodrigo Freitas
Abstract:
Metallic alloys often form phases - known as solid solutions - in which chemical elements are spread out on the same crystal lattice in an almost random manner. The tendency of certain chemical motifs to be more common than others is known as chemical short-range order (SRO) and it has received substantial consideration in alloys with multiple chemical elements present in large concentrations due…
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Metallic alloys often form phases - known as solid solutions - in which chemical elements are spread out on the same crystal lattice in an almost random manner. The tendency of certain chemical motifs to be more common than others is known as chemical short-range order (SRO) and it has received substantial consideration in alloys with multiple chemical elements present in large concentrations due to their extreme configurational complexity (e.g., high-entropy alloys). Short-range order renders solid solutions "slightly less random than completely random", which is a physically intuitive picture, but not easily quantifiable due to the sheer number of possible chemical motifs and their subtle spatial distribution on the lattice. Here we present a multiscale method to predict and quantify the SRO state of an alloy with atomic resolution, incorporating machine learning techniques to bridge the gap between electronic-structure calculations and the characteristic length scale of SRO. The result is an approach capable of predicting SRO length scale in agreement with experimental measurements while comprehensively correlating SRO with fundamental quantities such as local lattice distortions. This work advances the quantitative understanding of solid-solution phases, paving the way for SRO rigorous incorporation into predictive mechanical and thermodynamic models.
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Submitted 19 June, 2024; v1 submitted 2 November, 2023;
originally announced November 2023.
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Cracking the Quantum Scaling Limit with Machine Learned Electron Densities
Authors:
Joshua A. Rackers,
Lucas Tecot,
Mario Geiger,
Tess E. Smidt
Abstract:
A long-standing goal of science is to accurately solve the Schrödinger equation for large molecular systems. The poor scaling of current quantum chemistry algorithms on classical computers imposes an effective limit of about a few dozen atoms for which we can calculate molecular electronic structure. We present a machine learning (ML) method to break through this scaling limit and make quantum che…
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A long-standing goal of science is to accurately solve the Schrödinger equation for large molecular systems. The poor scaling of current quantum chemistry algorithms on classical computers imposes an effective limit of about a few dozen atoms for which we can calculate molecular electronic structure. We present a machine learning (ML) method to break through this scaling limit and make quantum chemistry calculations of very large systems possible. We show that Euclidean Neural Networks can be trained to predict the electron density with high fidelity from limited data. Learning the electron density allows us to train a machine learning model on small systems and make accurate predictions on large ones. We show that this ML electron density model can break through the quantum scaling limit and calculate the electron density of systems of thousands of atoms with quantum accuracy.
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Submitted 10 February, 2022; v1 submitted 10 January, 2022;
originally announced January 2022.
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Machine Learning on Neutron and X-Ray Scattering
Authors:
Zhantao Chen,
Nina Andrejevic,
Nathan Drucker,
Thanh Nguyen,
R Patrick Xian,
Tess Smidt,
Yao Wang,
Ralph Ernstorfer,
Alan Tennant,
Maria Chan,
Mingda Li
Abstract:
Neutron and X-ray scattering represent two state-of-the-art materials characterization techniques that measure materials' structural and dynamical properties with high precision. These techniques play critical roles in understanding a wide variety of materials systems, from catalysis to polymers, nanomaterials to macromolecules, and energy materials to quantum materials. In recent years, neutron a…
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Neutron and X-ray scattering represent two state-of-the-art materials characterization techniques that measure materials' structural and dynamical properties with high precision. These techniques play critical roles in understanding a wide variety of materials systems, from catalysis to polymers, nanomaterials to macromolecules, and energy materials to quantum materials. In recent years, neutron and X-ray scattering have received a significant boost due to the development and increased application of machine learning to materials problems. This article reviews the recent progress in applying machine learning techniques to augment various neutron and X-ray scattering techniques. We highlight the integration of machine learning methods into the typical workflow of scattering experiments. We focus on scattering problems that faced challenge with traditional methods but addressable using machine learning, such as leveraging the knowledge of simple materials to model more complicated systems, learning with limited data or incomplete labels, identifying meaningful spectra and materials' representations for learning tasks, mitigating spectral noise, and many others. We present an outlook on a few emerging roles machine learning may play in broad types of scattering and spectroscopic problems in the foreseeable future.
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Submitted 5 February, 2021;
originally announced February 2021.
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E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials
Authors:
Simon Batzner,
Albert Musaelian,
Lixin Sun,
Mario Geiger,
Jonathan P. Mailoa,
Mordechai Kornbluth,
Nicola Molinari,
Tess E. Smidt,
Boris Kozinsky
Abstract:
This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, res…
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This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, resulting in a more information-rich and faithful representation of atomic environments. The method achieves state-of-the-art accuracy on a challenging and diverse set of molecules and materials while exhibiting remarkable data efficiency. NequIP outperforms existing models with up to three orders of magnitude fewer training data, challenging the widely held belief that deep neural networks require massive training sets. The high data efficiency of the method allows for the construction of accurate potentials using high-order quantum chemical level of theory as reference and enables high-fidelity molecular dynamics simulations over long time scales.
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Submitted 16 December, 2021; v1 submitted 8 January, 2021;
originally announced January 2021.
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Direct prediction of phonon density of states with Euclidean neural networks
Authors:
Zhantao Chen,
Nina Andrejevic,
Tess Smidt,
Zhiwei Ding,
Yen-Ting Chi,
Quynh T. Nguyen,
Ahmet Alatas,
Jing Kong,
Mingda Li
Abstract:
Machine learning has demonstrated great power in materials design, discovery, and property prediction. However, despite the success of machine learning in predicting discrete properties, challenges remain for continuous property prediction. The challenge is aggravated in crystalline solids due to crystallographic symmetry considerations and data scarcity. Here we demonstrate the direct prediction…
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Machine learning has demonstrated great power in materials design, discovery, and property prediction. However, despite the success of machine learning in predicting discrete properties, challenges remain for continuous property prediction. The challenge is aggravated in crystalline solids due to crystallographic symmetry considerations and data scarcity. Here we demonstrate the direct prediction of phonon density of states using only atomic species and positions as input. We apply Euclidean neural networks, which by construction are equivariant to 3D rotations, translations, and inversion and thereby capture full crystal symmetry, and achieve high-quality prediction using a small training set of $\sim 10^{3}$ examples with over 64 atom types. Our predictive model reproduces key features of experimental data and even generalizes to materials with unseen elements,and is naturally suited to efficiently predict alloy systems without additional computational cost. We demonstrate the potential of our network by predicting a broad number of high phononic specific heat capacity materials. Our work indicates an efficient approach to explore materials' phonon structure, and can further enable rapid screening for high-performance thermal storage materials and phonon-mediated superconductors.
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Submitted 2 February, 2021; v1 submitted 10 September, 2020;
originally announced September 2020.
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Finding Symmetry Breaking Order Parameters with Euclidean Neural Networks
Authors:
Tess E. Smidt,
Mario Geiger,
Benjamin Kurt Miller
Abstract:
Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them". We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions into simple optimization problems. We prove these properties mathematically and demonstrate them numerically…
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Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them". We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions into simple optimization problems. We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry-breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites.
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Submitted 26 October, 2020; v1 submitted 4 July, 2020;
originally announced July 2020.
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A new spin-anisotropic harmonic honeycomb iridate
Authors:
Kimberly A. Modic,
Tess E. Smidt,
Itamar Kimchi,
Nicholas P. Breznay,
Alun Biffin,
Sungkyun Choi,
Roger D. Johnson,
Radu Coldea,
Pilanda Watkins-Curry,
Gregory T. McCandless,
Felipe Gandara,
Z. Islam,
Ashvin Vishwanath,
Julia Y. Chan,
Arkady Shekhter,
Ross D. McDonald,
James G. Analytis
Abstract:
The physics of Mott insulators underlies diverse phenomena ranging from high temperature superconductivity to exotic magnetism. Although both the electron spin and the structure of the local orbitals play a key role in this physics, in most systems these are connected only indirectly --- via the Pauli exclusion principle and the Coulomb interaction. Iridium-based oxides (iridates) open a further d…
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The physics of Mott insulators underlies diverse phenomena ranging from high temperature superconductivity to exotic magnetism. Although both the electron spin and the structure of the local orbitals play a key role in this physics, in most systems these are connected only indirectly --- via the Pauli exclusion principle and the Coulomb interaction. Iridium-based oxides (iridates) open a further dimension to this problem by introducing strong spin-orbit interactions, such that the Mott physics has a strong orbital character. In the layered honeycomb iridates this is thought to generate highly spin-anisotropic interactions, coupling the spin orientation to a given spatial direction of exchange and leading to strongly frustrated magnetism. The potential for new physics emerging from such interactions has driven much scientific excitement, most recently in the search for a new quantum spin liquid, first discussed by Kitaev \cite{kitaev_anyons_2006}. Here we report a new iridate structure that has the same local connectivity as the layered honeycomb, but in a three-dimensional framework. The temperature dependence of the magnetic susceptibility exhibits a striking reordering of the magnetic anisotropy, giving evidence for highly spin-anisotropic exchange interactions. Furthermore, the basic structural units of this material suggest the possibility of a new family of structures, the `harmonic honeycomb' iridates. This compound thus provides a unique and exciting glimpse into the physics of a new class of strongly spin-orbit coupled Mott insulators.
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Submitted 20 August, 2014; v1 submitted 13 February, 2014;
originally announced February 2014.