mlsanityData to 7 Oct 2026

Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

arXiv 2402.10926 ↗First posted 30 Jan 2024math.NA cs.LG cs.NAcited by 39

Not on the map. Why?

Abstract (arXiv)

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed machine learning. We provide a unified framework in which analysis of the various components of the error incurred by PINNs in approximating PDEs can be effectively carried out. A detailed review of available results on approximation, generalization and training errors and their behavior with respect to the type of the PDE and the dimension of the underlying domain is presented. In particular, the role of the regularity of the solutions and their stability to perturbations in the error analysis is elucidated. Numerical results are also presented to illustrate the theory. We identify training errors as a key bottleneck which can adversely affect the overall performance of various models in physics-informed machine learning.

Key facts. Cited by 39 papers in the map (data through 7 Oct 2026). Builds on 23 of its 108 references that are in the map. First posted on arXiv 30 Jan 2024.

Builds on

23 of 108 references.

References · 23

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  3. 2018On the Spectral Bias of Neural Networkscited by 1,321
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  6. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  7. 2014Breaking the Curse of Dimensionality with Convex Neural Networkscited by 329
  8. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  9. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  10. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  11. 2020On the linearity of large non-linear models: when and why the tangent kernel is constantcited by 129
  12. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
11 more
  1. 2020Representation formulas and pointwise properties for Barron functionscited by 62
  2. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  3. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  4. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  5. 2020On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamicscited by 39
  6. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  7. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  8. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  9. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  10. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 18
  11. 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 20; 2026: 18.

39 papers.

Sort:

  1. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  2. 2025Operator Learning at Machine Precisioncited by 6
  3. 2025A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networkscited by 5
  4. 2025Aspects of holographic entanglement using physics-informed-neural-networkscited by 5
  5. 2024First Order System Least Squares Neural Networkscited by 4
  6. 2025Prediction error certification for PINNs: Theory, computation, and application to Stokes flowcited by 4
  7. 2025Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEscited by 3
  8. 2026Learning, Solving and Optimizing PDEs with TensorGalerkin: an efficient high-performance Galerkin assembly algorithmcited by 3
  9. 2025HyperNOs: Automated and Parallel Library for Neural Operators Researchcited by 2
  10. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  11. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  12. 2025Expressive Power of Deep Networks on Manifolds: Simultaneous Approximationcited by 2
27 more
  1. 2025A data augmentation strategy for deep neural networks with application to epidemic modellingcited by 1
  2. 2025Langevin Monte-Carlo Provably Learns Depth Two Neural Nets at Any Size and Datacited by 1
  3. 2025Accelerating Natural Gradient Descent for PINNs with Randomized Numerical Linear Algebracited by 1
  4. 2025Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditionscited by 1
  5. 2025Augmented data and neural networks for robust epidemic forecasting: application to COVID-19 in Italycited by 1
  6. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  7. 2025Error analysis for the deep Kolmogorov methodcited by 0
  8. 2025THINNs: Thermodynamically Informed Neural Networkscited by 0
  9. 2025Domain-Informed Genetic Superposition Programming: A Case Study on SFRC Beamscited by 0
  10. 2025Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architecturescited by 0
  11. 2026Gauss-Newton Natural Gradient Descent for Shape Learningcited by 0
  12. 2026Symbolic recovery of PDEs from measurement datacited by 0
  13. 2026Building Trust in PINNs: Error Estimation through Finite Difference Methodscited by 0
  14. 2026A Family of Adaptive Activation Functions for Mitigating Failure Modes in Physics-Informed Neural Networkscited by 0
  15. 2026A Theory-guided Weighted L2 Loss for solving the BGK model via Physics-informed neural networkscited by 0
  16. 2026DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splittingcited by 0
  17. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0
  18. 2026Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observationscited by 0
  19. 2026Learning universal approximations for partial differential equations with Physics-Informed Broad Learning Systemcited by 0
  20. 2026Neural network surrogates with uncertainty quantification for inverse problems in partial differential equationscited by 0
  21. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  22. 2026The Differential Neural Tangent Kernel and Its Positivitycited by 0
  23. 2026Finite basis physics-informed neural networks with hard constraints for viscous fluid flow in highly perforated domainscited by 0
  24. 2026Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Lawscited by 0
  25. 2026Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyondcited by 0
  26. 2026Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selectioncited by 0
  27. 2026Transferability of Learned States in Neural PDE Solverscited by 0