mlsanityData to 7 Oct 2026

wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws

arXiv 2207.08483 ↗First posted 18 Jul 2022math.NA cs.LG cs.NAcited by 29

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Abstract (arXiv)

Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). Builds on 23 of its 45 references that are in the map. First posted on arXiv 18 Jul 2022.

Builds on

23 of 45 references.

References · 23

  1. 2022Generative Adversarial Networkscited by 5,603
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2017Wasserstein GANcited by 635
  4. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  5. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  6. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  7. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  8. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  10. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  11. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  12. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
11 more
  1. 2021On the approximation of functions by tanh neural networkscited by 81
  2. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  3. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  4. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  5. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  6. 2019Deep learning observables in computational fluid dynamicscited by 38
  7. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  8. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  9. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  10. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
  11. 2020Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequencescited by 20

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 6; 2024: 5; 2025: 6; 2026: 10.

29 papers.

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  1. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  2. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  3. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  4. 2022Variationally Mimetic Operator Networkscited by 19
  5. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  6. 2024GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Lawscited by 8
  7. 2023Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutionscited by 5
  8. 2024Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysiscited by 4
  9. 2024Self-test loss functions for learning weak-form operators and gradient flowscited by 4
  10. 2024Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEscited by 3
  11. 2026Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithmcited by 3
  12. 2023PINNs error estimates for nonlinear equations in ℝ-smooth Banach spacescited by 2
17 more
  1. 2023Conservative Physics-Informed Neural Networks for Non-Conservative Hyperbolic Conservation Laws Near Critical Statescited by 1
  2. 2025Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebracited by 1
  3. 2025SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Lawscited by 1
  4. 2025A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Seriescited by 1
  5. 2023A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part Icited by 0
  6. 2023Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networkscited by 0
  7. 2025(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEscited by 0
  8. 2025THINNs: Thermodynamically Informed Neural Networkscited by 0
  9. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 0
  10. 2026Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstructioncited by 0
  11. 2026A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Lawscited by 0
  12. 2026GeoFunFlow-3D: A Physics-Guided Generative Flow Matching Framework for High-Fidelity 3D Aerodynamic Inference over Complex Geometriescited by 0
  13. 2026Learning Neural Operator Surrogates for the Black Hole Accretion Codecited by 0
  14. 2026Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidancecited by 0
  15. 2026Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problemcited by 0
  16. 2026Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Lawscited by 0
  17. 2026Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Lawscited by 0