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Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs

arXiv 2601.06388 ↗First posted 10 Jan 2026math.NA cs.LG cs.NAcited by 0

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Abstract (arXiv)

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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References · 22

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
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  5. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
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  7. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  8. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  10. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  11. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  12. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
10 more
  1. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  2. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  3. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  4. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  5. 2023The Parametric Complexity of Operator Learningcited by 27
  6. 2018Nonlinear integro-differential operator regression with neural networkscited by 11
  7. 2024Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Lawscited by 10
  8. 2023Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errorscited by 6
  9. 2023Operator learning for hyperbolic partial differential equationscited by 4
  10. 2022Decentralized Vehicle Coordination: The Berkeley DeepDrive Drone Dataset and Consensus-Based Modelscited by 3

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