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

arXiv: https://arxiv.org/abs/2207.08483 · HTML: https://mlsanity.com/paper/2207.08483/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 45 references that are in the map.
- First posted on arXiv 18 Jul 2022.

## Builds on (most cited first, 23 of 23)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs](https://mlsanity.com/paper/2004.01806/) (2020)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences](https://mlsanity.com/paper/2005.12564/) (2020)

## Built on by (most cited first, 29 of 29)

- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities](https://mlsanity.com/paper/2210.01074/) (2022)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions](https://mlsanity.com/paper/2310.19590/) (2023)
- [Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis](https://mlsanity.com/paper/2409.20383/) (2024)
- [Self-test loss functions for learning weak-form operators and gradient flows](https://mlsanity.com/paper/2412.03506/) (2024)
- [Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs](https://mlsanity.com/paper/2404.17868/) (2024)
- [Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithm](https://mlsanity.com/paper/2602.05052/) (2026)
- [PINNs error estimates for nonlinear equations in ℝ-smooth Banach spaces](https://mlsanity.com/paper/2305.11915/) (2023)
- [Conservative Physics-Informed Neural Networks for Non-Conservative Hyperbolic Conservation Laws Near Critical States](https://mlsanity.com/paper/2305.12817/) (2023)
- [Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebra](https://mlsanity.com/paper/2505.11638/) (2025)
- [SPIKE: Stable Physics-Informed Kernel Evolution Method for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2510.18266/) (2025)
- [A Weak Penalty Neural ODE for Learning Chaotic Dynamics from Noisy Time Series](https://mlsanity.com/paper/2511.06609/) (2025)
- [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026)
- [A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I](https://mlsanity.com/paper/2307.04121/) (2023)
- [Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networks](https://mlsanity.com/paper/2307.06362/) (2023)
- [(U)NFV: Supervised and Unsupervised Neural Finite Volume Methods for Solving Hyperbolic PDEs](https://mlsanity.com/paper/2505.23702/) (2025)
- [THINNs: Thermodynamically Informed Neural Networks](https://mlsanity.com/paper/2509.19467/) (2025)
- [Curriculum-Learned Vanishing Stacked Residual PINNs for Hyperbolic PDE State Reconstruction](https://mlsanity.com/paper/2602.06996/) (2026)
- [A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws](https://mlsanity.com/paper/2604.15617/) (2026)
- [GeoFunFlow-3D: A Physics-Guided Generative Flow Matching Framework for High-Fidelity 3D Aerodynamic Inference over Complex Geometries](https://mlsanity.com/paper/2604.23350/) (2026)
- [Learning Neural Operator Surrogates for the Black Hole Accretion Code](https://mlsanity.com/paper/2604.25985/) (2026)
- [Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance](https://mlsanity.com/paper/2605.11001/) (2026)
- [Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem](https://mlsanity.com/paper/2607.27433/) (2026)
- [Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws](https://mlsanity.com/paper/2608.10389/) (2026)
- [Neural Network-Assisted Refinement of Traditional Schemes for One-Dimensional Scalar Conservation Laws](https://mlsanity.com/paper/2609.32887/) (2026)

## 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.