mlsanityData to 7 Oct 2026

Can Physics-Informed Neural Networks beat the Finite Element Method?

arXiv 2302.04107 ↗First posted 8 Feb 2023math.NA cs.LG cs.NAcited by 68

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Abstract (arXiv)

Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen-Cahn in 1D, semilinear Schr"odinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

Key facts. Cited by 68 papers in the map (data through 7 Oct 2026). Builds on 26 of its 84 references that are in the map. First posted on arXiv 8 Feb 2023.

Builds on

26 of 84 references.

References · 26

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  7. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
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  9. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
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  11. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
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14 more
  1. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  2. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  3. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  4. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
  5. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  6. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
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  10. 2021Conditional physics informed neural networkscited by 23
  11. 2020Can Shallow Neural Networks Beat the Curse of Dimensionality? A mean field training perspectivecited by 22
  12. 2022Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustrationcited by 21
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 13; 2024: 15; 2025: 26; 2026: 14.

68 papers.

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  1. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  2. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  3. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 84
  4. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  5. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  6. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  7. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  8. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  9. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  10. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 10
  11. 2023Multilevel CNNs for Parametric PDEscited by 9
  12. 2024A Learnable Prior Improves Inverse Tumor Growth Modelingcited by 9
38 more
  1. 2025Solving Sparse Finite Element Problems on Neuromorphic Hardwarecited by 8
  2. 2024ANaGRAM: A Natural Gradient Relative to Adapted Model for efficient PINNs learningcited by 7
  3. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 6
  4. 2023PINN surrogate of Li-ion battery models for parameter inference. Part I: Implementation and multi-fidelity hierarchies for the single-particle modelcited by 6
  5. 2024Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equationscited by 6
  6. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  7. 2025A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometriescited by 5
  8. 2024Bayesian inverse Navier-Stokes problems: joint flow field reconstruction and parameter learningcited by 4
  9. 2024Response Estimation and System Identification of Dynamical Systems via Physics-Informed Neural Networkscited by 4
  10. 2024Symplectic Neural Flows for Modeling and Discoverycited by 4
  11. 2025Prediction error certification for PINNs: Theory, computation, and application to Stokes flowcited by 4
  12. 2024Physics-Guided Neural Networks for Intraventricular Vector Flow Mappingcited by 3
  13. 2024Solving Differential Equations with Constrained Learningcited by 3
  14. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  15. 2025LNN-PINN: A Unified Physics-Only Training Framework with Liquid Residual Blockscited by 3
  16. 2023TSONN: Time-stepping-oriented neural network for solving partial differential equationscited by 2
  17. 2024Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equationscited by 2
  18. 2024Sobolev neural network with residual weighting as a surrogate in linear and non-linear mechanicscited by 2
  19. 2024SetPINNs: Set-based Physics-informed Neural Networkscited by 2
  20. 2025Generalized Lie Symmetries in Physics-Informed Neural Operatorscited by 2
  21. 2025Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptationcited by 2
  22. 2025MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Accelerationcited by 2
  23. 2025Homogenization with Guaranteed Bounds via Primal-Dual Physically Informed Neural Networkscited by 2
  24. 2025Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problemscited by 2
  25. 2023GPINN: Physics-informed Neural Network with Graph Embeddingcited by 1
  26. 2024Improved physics-informed neural network in mitigating gradient related failurescited by 1
  27. 2025BO-SA-PINNs: Self-adaptive physics-informed neural networks based on Bayesian optimization for automatically designing PDE solverscited by 1
  28. 2025Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equationscited by 1
  29. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  30. 2026Equation Recast for Canonical Operator Learning Across Parametric PDEscited by 1
  31. 2023Spectral-Bias and Kernel-Task Alignment in Physically Informed Neural Networkscited by 0
  32. 2024Science-Informed Design of Deep Learning With Applications to Wireless Systems: A Tutorialcited by 0
  33. 2025Differential equation quantum solvers: engineering measurements to reduce costcited by 0
  34. 2025OmniFluids: Physics Pre-trained Modeling of Fluid Dynamicscited by 0
  35. 2025Operator Learning for Consolidation: An Architectural Comparison for DeepONet Variantscited by 0
  36. 2025Improved Training Strategies for Physics-Informed Neural Networks using Real Experimental Data in Aluminum Spot Weldingcited by 0
  37. 2025One-shot acceleration of transient PDE solvers via online-learned preconditionerscited by 0
  38. 2025Benchmarking Physics-Informed Neural Networks and Boundary Elements Methods for Wave Scatteringcited by 0

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