mlsanityData to 14 Oct 2026

Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study

arXiv 2503.17379 ↗First posted 14 Mar 2025physics.flu-dyn cs.LGcited by 11

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Abstract (arXiv)

Solving partial differential equations (PDEs) with discontinuous solutions , such as shock waves in multiphase viscous flow in porous media , is critical for a wide range of scientific and engineering applications, as they represent sudden changes in physical quantities. Physics-Informed Neural Networks (PINNs), an approach proposed for solving PDEs, encounter significant challenges when applied to such systems. Accurately solving PDEs with discontinuities using PINNs requires specialized techniques to ensure effective solution accuracy and numerical stability. A benchmarking study was conducted on two multiphase flow problems in porous media: the classic Buckley-Leverett (BL) problem and a fully coupled system of equations involving shock waves but with varying levels of solution complexity. The findings show that PM and LM approaches can provide accurate solutions for the BL problem by effectively addressing the infinite gradients associated with shock occurrences. In contrast, AM methods failed to effectively resolve the shock waves. When applied to fully coupled PDEs (with more complex loss landscape), the generalization error in the solutions quickly increased, highlighting the need for ongoing innovation. This study provides a comprehensive review of existing techniques for managing PDE discontinuities using PINNs, offering information on their strengths and limitations. The results underscore the necessity for further research to improve PINNs ability to handle complex discontinuities, particularly in more challenging problems with complex loss landscapes. This includes problems involving higher dimensions or multiphysics systems, where current methods often struggle to maintain accuracy and efficiency.

Key facts. Cited by 11 papers in the map (data through 14 Oct 2026). Builds on 27 of its 98 references that are in the map. First posted on arXiv 14 Mar 2025.

Builds on

27 of 98 references.

References · 27

  1. 2017Attention Is All You Needcited by 60,361
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  3. 2018On the Spectral Bias of Neural Networkscited by 1,341
  4. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  5. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  6. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  7. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  8. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  9. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 76
  10. 2023Residual-based attention in physics-informed neural networkscited by 72
  11. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  12. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
15 more
  1. 2021Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functionscited by 49
  2. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  3. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  4. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  5. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  6. 2022Physics-Informed Neural Networks with Adaptive Localized Artificial Viscositycited by 22
  7. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  8. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
  9. 2022Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)cited by 10
  10. 2024Binary structured physics-informed neural networks for solving equations with rapidly changing solutionscited by 8
  11. 2024GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Lawscited by 8
  12. 2024Understanding the Difficulty of Solving Cauchy Problems with PINNscited by 6
  13. 2022Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problemscited by 5
  14. 2023Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDEcited by 2
  15. 2024History-Matching of Imbibition Flow in Multiscale Fractured Porous Media Using Physics-Informed Neural Networks (PINNs)cited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 4; 2026: 7.

11 papers.

  1. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  2. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  3. 2025BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamicscited by 5
  4. 2026FEKAN: Feature-Enriched Kolmogorov-Arnold Networkscited by 5
  5. 2026A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modelingcited by 2
  6. 2026Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradientscited by 2
  7. 2026In Transformer We Trust? A Perspective on Transformer Architecture Failure Modescited by 1
  8. 2026GenVoid: Uncertainty-Aware Learning of Subsurface Material Defects with an Experimentally Validated Physics-Informed Generative Modelcited by 1
  9. 2025Spatio-temporal, multi-field deep learning of shock propagation in meso-structured mediacited by 0
  10. 2026Soft Partition-based KAPI-ELM for Multi-Scale PDEscited by 0
  11. 2026Learning Neural Operator Surrogates for the Black Hole Accretion Codecited by 0