mlsanityData to 7 Oct 2026

Error estimates for physics informed neural networks approximating the Navier-Stokes equations

arXiv 2203.09346 ↗First posted 17 Mar 2022math.NA cs.LG cs.NAcited by 63

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Abstract (arXiv)

We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.

Key facts. Cited by 63 papers in the map (data through 7 Oct 2026). Builds on 20 of its 54 references that are in the map. First posted on arXiv 17 Mar 2022.

Builds on

20 of 54 references.

References · 20

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  3. 2018Stronger generalization bounds for deep nets via a compression approachcited by 306
  4. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  5. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  6. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  7. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  8. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  9. 2021Long-time integration of parametric evolution equations with physics-informed DeepONetscited by 87
  10. 2021On the approximation of functions by tanh neural networkscited by 81
  11. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  12. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
8 more
  1. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  2. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  3. 2019Deep learning observables in computational fluid dynamicscited by 38
  4. 2018Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Datacited by 36
  5. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  6. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  7. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
  8. 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: 7; 2023: 19; 2024: 13; 2025: 10; 2026: 14.

63 papers.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  3. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  4. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  5. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 43
  6. 2024Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?cited by 31
  7. 2022How important are activation functions in regression and classification? A survey, performance comparison, and future directionscited by 29
  8. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  9. 2023Recent Developments in Machine Learning Methods for Stochastic Control and Gamescited by 19
  10. 2023MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networkscited by 17
  11. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  12. 2024Preconditioning for Physics-Informed Neural Networkscited by 12
38 more
  1. 2025Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Studycited by 11
  2. 2023On the Generalization of PINNs outside the training domain and the Hyperparameters influencing itcited by 10
  3. 2024GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applicationscited by 10
  4. 2022Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networkscited by 6
  5. 2023Symbolic Regression for PDEs using Pruned Differentiable Programscited by 6
  6. 2024Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equationscited by 6
  7. 2023Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performancecited by 5
  8. 2023Entropy-dissipation Informed Neural Network for McKean-Vlasov Type PDEscited by 5
  9. 2023Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networkscited by 5
  10. 2024Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous mediacited by 5
  11. 2024Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networkscited by 5
  12. 2024Neural Operators Can Play Dynamic Stackelberg Gamescited by 5
  13. 2025BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamicscited by 5
  14. 2025Higher Order Approximation Rates for ReLU CNNs in Korobov Spacescited by 4
  15. 2025Prediction error certification for PINNs: Theory, computation, and application to Stokes flowcited by 4
  16. 2022Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEscited by 3
  17. 2025Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundlescited by 3
  18. 2022Error analysis for deep neural network approximations of parametric hyperbolic conservation lawscited by 2
  19. 2023Physics-aware deep learning framework for linear elasticitycited by 2
  20. 2023PINNs error estimates for nonlinear equations in ℝ-smooth Banach spacescited by 2
  21. 2023Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUpcited by 2
  22. 2024Exact and approximate error bounds for physics-informed neural networkscited by 2
  23. 2025Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networkscited by 2
  24. 2023Error Analysis of Physics-Informed Neural Networks for Approximating Dynamic PDEs of Second Order in Timecited by 1
  25. 2023Deep smoothness WENO scheme for two-dimensional hyperbolic conservation laws: A deep learning approach for learning smoothness indicatorscited by 1
  26. 2024Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equationscited by 1
  27. 2024Astral: training physics-informed neural networks with error majorantscited by 1
  28. 2024Error Bounds for Physics-Informed Neural Networks in Fokker-Planck PDEscited by 1
  29. 2026AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Trainingcited by 1
  30. 2023Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networkscited by 0
  31. 2023Investigation of Compressor Cascade Flow Using Physics- Informed Neural Networks with Adaptive Learning Strategycited by 0
  32. 2023A numerical approach for the fractional Laplacian via deep neural networkscited by 0
  33. 2023Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zonescited by 0
  34. 2024Error Analysis and Numerical Algorithm for PDE Approximation with Hidden-Layer Concatenated Physics Informed Neural Networkscited by 0
  35. 2025Provably Accurate Adaptive Sampling for Collocation Points in Physics-informed Neural Networkscited by 0
  36. 2025M'ethode de quadrature pour les PINNs fond'ee th'eoriquement sur la hessienne des r'esiduelscited by 0
  37. 2025THINNs: Thermodynamically Informed Neural Networkscited by 0
  38. 2025Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Methodcited by 0

50 most cited of 63 All 63 (JSON)