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HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws
Authors:
Dimitrije Ždrale,
Cassie An Jeng,
Katie Wang,
Sonia Vanier,
Alexandre Bayen,
Hossein Nick Zinat Matin
Abstract:
We introduce HypNO, a graph-based neural operator for scalar hyperbolic conservation laws. HypNO operates directly on a space-time graph of finite-volume cells and uses adjacency-factored, physics-informed message passing to respect upwinding and entropy admissibility near shocks. We benchmark the architecture on the Lighthill-Whitham-Richards (LWR) and Aw-Rascle-Zhang (ARZ) traffic-flow models, a…
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We introduce HypNO, a graph-based neural operator for scalar hyperbolic conservation laws. HypNO operates directly on a space-time graph of finite-volume cells and uses adjacency-factored, physics-informed message passing to respect upwinding and entropy admissibility near shocks. We benchmark the architecture on the Lighthill-Whitham-Richards (LWR) and Aw-Rascle-Zhang (ARZ) traffic-flow models, a stress test for operator-learning methods because of their simultaneous global transport and shock formation. HypNO predicts solution snapshots accurately across a range of initial conditions while capturing the shocks and discontinuities of the solution.
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Submitted 10 July, 2026;
originally announced July 2026.
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Existence and uniqueness of nonlocal nonlinear conservation laws via fixed-point methods
Authors:
Xiaoqian Gong,
Alexander Keimer,
Lorenzo Liverani,
Hossein Nick Zinat Matin
Abstract:
We investigate the well-posedness of scalar conservation laws whose flux depends on the solution both pointwise and nonlocally through integral averages. Our analysis is based on a fixed-point formulation, in which the nonlocal dependence is incorporated as a space- and time-dependent component of the flux, together with classical stability estimates for entropy solutions. This framework unifies a…
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We investigate the well-posedness of scalar conservation laws whose flux depends on the solution both pointwise and nonlocally through integral averages. Our analysis is based on a fixed-point formulation, in which the nonlocal dependence is incorporated as a space- and time-dependent component of the flux, together with classical stability estimates for entropy solutions. This framework unifies and extends several models previously considered in the literature and applies, in particular, to conservation laws with memory effects (nonlocality in time) or delay. We prove the existence and uniqueness of weak entropy solutions on a sufficiently short time horizon and show that under additional assumptions, existence and uniqueness can be obtained on any finite time horizon. In addition, we present numerical simulations to illustrate the qualitative effects of memory on the solution dynamics.
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Submitted 9 April, 2026;
originally announced April 2026.
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Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs
Authors:
Zakaria Baba,
Alexandre M. Bayen,
Alexi Canesse,
Maria Laura Delle Monache,
Martin Drieux,
Zhe Fu,
Nathan Lichtlé,
Zihe Liu,
Hossein Nick Zinat Matin,
Benedetto Piccoli
Abstract:
We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative structure of the scheme. The model can be trained both in a supervised setting with efficiently generated synthetic data or in an unsupervised manner, leveraging the w…
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We present a neural network-based method for learning scalar hyperbolic conservation laws. Our method replaces the traditional numerical flux in finite volume schemes with a trainable neural network while preserving the conservative structure of the scheme. The model can be trained both in a supervised setting with efficiently generated synthetic data or in an unsupervised manner, leveraging the weak formulation of the partial differential equation. We provide theoretical results that our model can perform arbitrarily well, and provide associated upper bounds on neural network size. Extensive experiments demonstrate that our method often outperforms efficient schemes such as Godunov's scheme, WENO, and Discontinuous Galerkin for comparable computational budgets. Finally, we demonstrate the effectiveness of our method on a traffic prediction task, leveraging field experimental highway data from the Berkeley DeepDrive drone dataset.
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Submitted 9 January, 2026;
originally announced January 2026.
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(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs
Authors:
Nathan Lichtlé,
Alexi Canesse,
Zhe Fu,
Hossein Nick Zinat Matin,
Maria Laura Delle Monache,
Alexandre M. Bayen
Abstract:
We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: conve…
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We introduce (U)NFV, a modular neural network architecture that generalizes classical finite volume (FV) methods for solving hyperbolic conservation laws. Hyperbolic partial differential equations (PDEs) are challenging to solve, particularly conservation laws whose physically relevant solutions contain shocks and discontinuities. FV methods are widely used for their mathematical properties: convergence to entropy solutions, flow conservation, or total variation diminishing, but often lack accuracy and flexibility in complex settings. Neural Finite Volume addresses these limitations by learning update rules over extended spatial and temporal stencils while preserving conservation structure. It supports both supervised training on solution data (NFV) and unsupervised training via weak-form residual loss (UNFV). Applied to first-order conservation laws, (U)NFV achieves up to 10x lower error than Godunov's method, outperforms ENO/WENO, and rivals discontinuous Galerkin solvers with far less complexity. On traffic modeling problems, both from PDEs and from experimental highway data, (U)NFV captures nonlinear wave dynamics with significantly higher fidelity and scalability than traditional FV approaches.
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Submitted 29 May, 2025;
originally announced May 2025.
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A nonlocal degenerate macroscopic model of traffic dynamics with saturated diffusion: modeling and calibration theory
Authors:
Dawson Do,
Hossein Nick Zinat Matin,
Masuma Mollika Miti,
Maria Laura Delle Monache
Abstract:
In this work, we introduce a novel first-order nonlocal partial differential equation with saturated diffusion to describe the macroscopic behavior of traffic dynamics. We show how the proposed model is better in comparison with existing models in explaining the underlying driver behavior in real traffic data. In doing so, we introduce a methodology for adjusting the parameters of the proposed PDE…
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In this work, we introduce a novel first-order nonlocal partial differential equation with saturated diffusion to describe the macroscopic behavior of traffic dynamics. We show how the proposed model is better in comparison with existing models in explaining the underlying driver behavior in real traffic data. In doing so, we introduce a methodology for adjusting the parameters of the proposed PDE with respect to the distribution of real datasets. In particular, we conceptually and analytically elaborate on how such calibration connects the solution of the PDE to the probability transition kernel proposed by the datasets.
The performance of the model is thoroughly investigated with respect to several metrics. More precisely, we study the capability of the model in capturing the probability distribution realized by the datasets in the form of the fundamental diagram. We show that the model is capable of approximating the dynamics of the evolution of the probability distribution. To this end, we evaluate the performance of the model with regard to the congestion formation and dissipation scenarios from various datasets.
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Submitted 20 March, 2025;
originally announced March 2025.
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Second-Order Time to Collision With Non-Static Acceleration
Authors:
Hossein Nick Zinat Matin,
Yuneil Yeo,
Amelie Ju-Kang Ngo,
Antonio R. Paiva,
Jean Utke,
Maria Laura Delle Monache
Abstract:
We propose a second-order time to collision (TTC) considering non-static acceleration and turning with realistic assumptions. This is equivalent to considering that the steering wheel is held at a fixed angle with constant pressure on the gas or brake pedal and matches the well-known bicycle model. Past works that use acceleration to compute TTC consider only longitudinally aligned acceleration.…
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We propose a second-order time to collision (TTC) considering non-static acceleration and turning with realistic assumptions. This is equivalent to considering that the steering wheel is held at a fixed angle with constant pressure on the gas or brake pedal and matches the well-known bicycle model. Past works that use acceleration to compute TTC consider only longitudinally aligned acceleration.
We additionally develop and present the Second-Order Time-to-Collision Algorithm using Region-based search (STAR) to efficiently compute the proposed second-order TTC and overcome the current limitations of the existing built-in functions. The evaluation of the algorithm in terms of error and computation time is conducted through statistical analysis.
Through numerical simulations and publicly accessible real-world trajectory datasets, we show that the proposed second-order TTC with non-static acceleration is superior at reflecting accurate collision times, especially when turning is involved.
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Submitted 11 February, 2025;
originally announced February 2025.
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Reinforcement learning-based adaptive speed controllers in mixed autonomy condition
Authors:
Han Wang,
Hossein Nick Zinat Matin,
Maria Laura Delle Monache
Abstract:
The integration of Automated Vehicles (AVs) into traffic flow holds the potential to significantly improve traffic congestion by enabling AVs to function as actuators within the flow. This paper introduces an adaptive speed controller tailored for scenarios of mixed autonomy, where AVs interact with human-driven vehicles. We model the traffic dynamics using a system of strongly coupled Partial and…
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The integration of Automated Vehicles (AVs) into traffic flow holds the potential to significantly improve traffic congestion by enabling AVs to function as actuators within the flow. This paper introduces an adaptive speed controller tailored for scenarios of mixed autonomy, where AVs interact with human-driven vehicles. We model the traffic dynamics using a system of strongly coupled Partial and Ordinary Differential Equations (PDE-ODE), with the PDE capturing the general flow of human-driven traffic and the ODE characterizing the trajectory of the AVs. A speed policy for AVs is derived using a Reinforcement Learning (RL) algorithm structured within an Actor-Critic (AC) framework. This algorithm interacts with the PDE-ODE model to optimize the AV control policy. Numerical simulations are presented to demonstrate the controller's impact on traffic patterns, showing the potential of AVs to improve traffic flow and reduce congestion.
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Submitted 17 August, 2024;
originally announced August 2024.
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On the Analytical Properties of a Nonlinear Microscopic Dynamical Model for Connected and Automated Vehicles
Authors:
H. Nick Zinat Matin,
Y. Yeo,
X. Gong,
M. L. Delle Monache
Abstract:
In this paper, we propose an integrated dynamical model of Connected and Automated Vehicles (CAVs) which incorporates CAV technologies and a microscopic car-following model to improve safety, efficiency and convenience. We rigorously investigate the analytical properties such as well-posedness, maximum principle, perturbation and stability of the proposed model in some proper functional spaces. Fu…
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In this paper, we propose an integrated dynamical model of Connected and Automated Vehicles (CAVs) which incorporates CAV technologies and a microscopic car-following model to improve safety, efficiency and convenience. We rigorously investigate the analytical properties such as well-posedness, maximum principle, perturbation and stability of the proposed model in some proper functional spaces. Furthermore, we prove that the model is collision free and we derive and explicit lower bound on the distance as a safety measure.
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Submitted 27 May, 2024;
originally announced May 2024.
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Traffic Control via Connected and Automated Vehicles: An Open-Road Field Experiment with 100 CAVs
Authors:
Jonathan W. Lee,
Han Wang,
Kathy Jang,
Amaury Hayat,
Matthew Bunting,
Arwa Alanqary,
William Barbour,
Zhe Fu,
Xiaoqian Gong,
George Gunter,
Sharon Hornstein,
Abdul Rahman Kreidieh,
Nathan Lichtlé,
Matthew W. Nice,
William A. Richardson,
Adit Shah,
Eugene Vinitsky,
Fangyu Wu,
Shengquan Xiang,
Sulaiman Almatrudi,
Fahd Althukair,
Rahul Bhadani,
Joy Carpio,
Raphael Chekroun,
Eric Cheng
, et al. (39 additional authors not shown)
Abstract:
The CIRCLES project aims to reduce instabilities in traffic flow, which are naturally occurring phenomena due to human driving behavior. These "phantom jams" or "stop-and-go waves,"are a significant source of wasted energy. Toward this goal, the CIRCLES project designed a control system referred to as the MegaController by the CIRCLES team, that could be deployed in real traffic. Our field experim…
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The CIRCLES project aims to reduce instabilities in traffic flow, which are naturally occurring phenomena due to human driving behavior. These "phantom jams" or "stop-and-go waves,"are a significant source of wasted energy. Toward this goal, the CIRCLES project designed a control system referred to as the MegaController by the CIRCLES team, that could be deployed in real traffic. Our field experiment leveraged a heterogeneous fleet of 100 longitudinally-controlled vehicles as Lagrangian traffic actuators, each of which ran a controller with the architecture described in this paper. The MegaController is a hierarchical control architecture, which consists of two main layers. The upper layer is called Speed Planner, and is a centralized optimal control algorithm. It assigns speed targets to the vehicles, conveyed through the LTE cellular network. The lower layer is a control layer, running on each vehicle. It performs local actuation by overriding the stock adaptive cruise controller, using the stock on-board sensors. The Speed Planner ingests live data feeds provided by third parties, as well as data from our own control vehicles, and uses both to perform the speed assignment. The architecture of the speed planner allows for modular use of standard control techniques, such as optimal control, model predictive control, kernel methods and others, including Deep RL, model predictive control and explicit controllers. Depending on the vehicle architecture, all onboard sensing data can be accessed by the local controllers, or only some. Control inputs vary across different automakers, with inputs ranging from torque or acceleration requests for some cars, and electronic selection of ACC set points in others. The proposed architecture allows for the combination of all possible settings proposed above. Most configurations were tested throughout the ramp up to the MegaVandertest.
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Submitted 26 February, 2024;
originally announced February 2024.
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Hierarchical Speed Planner for Automated Vehicles: A Framework for Lagrangian Variable Speed Limit in Mixed Autonomy Traffic
Authors:
Han Wang,
Zhe Fu,
Jonathan Lee,
Hossein Nick Zinat Matin,
Arwa Alanqary,
Daniel Urieli,
Sharon Hornstein,
Abdul Rahman Kreidieh,
Raphael Chekroun,
William Barbour,
William A. Richardson,
Dan Work,
Benedetto Piccoli,
Benjamin Seibold,
Jonathan Sprinkle,
Alexandre M. Bayen,
Maria Laura Delle Monache
Abstract:
This paper introduces a novel control framework for Lagrangian variable speed limits in hybrid traffic flow environments utilizing automated vehicles (AVs). The framework was validated using a fleet of 100 connected automated vehicles as part of the largest coordinated open-road test designed to smooth traffic flow. The framework includes two main components: a high-level controller deployed on th…
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This paper introduces a novel control framework for Lagrangian variable speed limits in hybrid traffic flow environments utilizing automated vehicles (AVs). The framework was validated using a fleet of 100 connected automated vehicles as part of the largest coordinated open-road test designed to smooth traffic flow. The framework includes two main components: a high-level controller deployed on the server side, named Speed Planner, and low-level controllers called vehicle controllers deployed on the vehicle side. The Speed Planner designs and updates target speeds for the vehicle controllers based on real-time Traffic State Estimation (TSE) [1]. The Speed Planner comprises two modules: a TSE enhancement module and a target speed design module. The TSE enhancement module is designed to minimize the effects of inherent latency in the received traffic information and to improve the spatial and temporal resolution of the input traffic data. The target speed design module generates target speed profiles with the goal of improving traffic flow. The vehicle controllers are designed to track the target speed meanwhile responding to the surrounding situation. The numerical simulation indicates the performance of the proposed method: the bottleneck throughput has increased by 5.01%, and the speed standard deviation has been reduced by a significant 34.36%. We further showcase an operational study with a description of how the controller was implemented on a field-test with 100 AVs and its comprehensive effects on the traffic flow.
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Submitted 26 February, 2024;
originally announced February 2024.
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High-Order Numerical Method for 1D Non-local Diffusive Equation
Authors:
D. Do,
H. Nick Zinat Matin,
M. L. Delle Monache
Abstract:
In this paper we present a non-local numerical scheme based on the Local Discontinuous Galerkin method for a non-local diffusive partial differential equation with application to traffic flow. In this model, the velocity is determined by both the average of the traffic density as well as the changes in the traffic density at a neighborhood of each point. We discuss nonphysical behaviors that can a…
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In this paper we present a non-local numerical scheme based on the Local Discontinuous Galerkin method for a non-local diffusive partial differential equation with application to traffic flow. In this model, the velocity is determined by both the average of the traffic density as well as the changes in the traffic density at a neighborhood of each point. We discuss nonphysical behaviors that can arise when including diffusion, and our measures to prevent them in our model. The numerical results suggest that this is an accurate method for solving this type of equation and that the model can capture desired traffic flow behavior. We show that computation of the non-local convolution results in $\mathcal{O}(n^2)$ complexity, but the increased computation time can be mitigated with high-order schemes like the one proposed.
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Submitted 11 November, 2023; v1 submitted 7 November, 2023;
originally announced November 2023.
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On the Existence of Solution of Conservation Law with Moving Bottleneck and Discontinuity in FLux
Authors:
Hossein Nick Zinat Matin,
Maria Laura Delle Monache
Abstract:
In this paper, a PDE-ODE model with discontinuity in the flux as well as a flux constraint is analyzed. A modified Riemann solution is proposed and the existence of a weak solution to the Cauchy problem is rigorously investigated using the wavefront tracking scheme.
In this paper, a PDE-ODE model with discontinuity in the flux as well as a flux constraint is analyzed. A modified Riemann solution is proposed and the existence of a weak solution to the Cauchy problem is rigorously investigated using the wavefront tracking scheme.
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Submitted 30 September, 2023;
originally announced October 2023.
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Near Collision and Controllability Analysis of Nonlinear Optimal Velocity Follow-the-Leader Dynamical Model In Traffic Flow
Authors:
Hossein Nick Zinat Matin,
Maria Laura Delle Monache
Abstract:
This paper examines the optimal velocity follow-the-leader dynamics, a microscopic traffic model, and explores different aspects of the dynamical model, with particular emphasis on collision analysis. More precisely, we present a rigorous boundary-layer analysis of the model which provides a careful understanding of the behavior of the dynamics in trade-off with the singularity of the model at col…
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This paper examines the optimal velocity follow-the-leader dynamics, a microscopic traffic model, and explores different aspects of the dynamical model, with particular emphasis on collision analysis. More precisely, we present a rigorous boundary-layer analysis of the model which provides a careful understanding of the behavior of the dynamics in trade-off with the singularity of the model at collision which is essential in the controllability of the system.
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Submitted 20 September, 2023;
originally announced September 2023.
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Near-collision dynamics in a noisy car-following model
Authors:
Hossein Nick Zinat Matin,
Richard B. Sowers
Abstract:
We consider a small stochastic perturbation of an optimal velocity car-following model. We give a detailed analysis of behavior near the collision singularity. We show that collision is impossible in a simplified model without noise, and then we show that collision is asymptotically unlikely over large time intervals in presence of small noise, with large time interval scaling like a square of the…
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We consider a small stochastic perturbation of an optimal velocity car-following model. We give a detailed analysis of behavior near the collision singularity. We show that collision is impossible in a simplified model without noise, and then we show that collision is asymptotically unlikely over large time intervals in presence of small noise, with large time interval scaling like a square of the reciprocal of the strength of the noise. Our calculations depend on careful boundary-layer analyses.
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Submitted 1 September, 2022; v1 submitted 2 November, 2021;
originally announced November 2021.