mlsanityData to 7 Oct 2026

Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

arXiv 2111.02801 ↗First posted 1 Nov 2021cs.LG physics.comp-phcited by 220

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Abstract (arXiv)

Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems. However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs), for improving the accuracy and training efficiency of PINNs. gPINNs leverage gradient information of the PDE residual and embed the gradient into the loss function. We tested gPINNs extensively and demonstrated the effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical results show that gPINN performs better than PINN with fewer training points. Furthermore, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the performance of gPINN, especially in PDEs with solutions that have steep gradients.

Key facts. Cited by 220 papers in the map (data through 7 Oct 2026). Builds on 20 of its 41 references that are in the map. First posted on arXiv 1 Nov 2021.

Builds on

20 of 41 references.

References · 20

  1. 2015Automatic differentiation in machine learning: a surveycited by 846
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  3. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  4. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  5. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  6. 2017Improving the Adversarial Robustness and Interpretability of Deep Neural Networks by Regularizing their Input Gradientscited by 236
  7. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  8. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  9. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  10. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
  11. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  12. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
8 more
  1. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 86
  2. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 78
  3. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 66
  4. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  5. 2019Scaleable input gradient regularization for adversarial robustnesscited by 45
  6. 2020PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometriescited by 38
  7. 2019A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problemscited by 34
  8. 2020Multifidelity Data Fusion via Gradient-Enhanced Gaussian Process Regressioncited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 31; 2023: 43; 2024: 52; 2025: 54; 2026: 39.

220 papers.

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  1. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  2. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  3. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
  4. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  5. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  6. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  7. 2023PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEscited by 84
  8. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  9. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  10. 2022Failure-informed adaptive sampling for PINNscited by 64
  11. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 55
  12. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
38 more
  1. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  2. 2023Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestrationcited by 46
  3. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 43
  4. 2024Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networkscited by 37
  5. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  6. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  7. 2022Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizationscited by 26
  8. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  9. 2023Physics-informed neural networks for solving forward and inverse problems in complex beam systemscited by 24
  10. 2023Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networkscited by 23
  11. 2022Scalable algorithms for physics-informed neural and graph networkscited by 21
  12. 2022Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problemscited by 21
  13. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  14. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  15. 2024Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operatorscited by 20
  16. 2023Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEscited by 18
  17. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 18
  18. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  19. 2022RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networkscited by 14
  20. 2023Learning Specialized Activation Functions for Physics-informed Neural Networkscited by 14
  21. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  22. 2025AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learningcited by 13
  23. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  24. 2022Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equationscited by 11
  25. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11
  26. 2023Turbulence in Focus: Benchmarking Scaling Behavior of 3D Volumetric Super-Resolution with BLASTNet 2.0 Datacited by 11
  27. 2024Functional Tensor Decompositions for Physics-Informed Neural Networkscited by 11
  28. 2024Efficient and generalizable nested Fourier-DeepONet for three-dimensional geological carbon sequestrationcited by 11
  29. 2025Lang-PINN: From Language to Physics-Informed Neural Networks via a Multi-Agent Frameworkcited by 11
  30. 2022Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spreadcited by 10
  31. 2025Sub-Sequential Physics-Informed Learning with State Space Modelcited by 10
  32. 2025Enabling Local Neural Operators to perform Equation-Free System-Level Analysiscited by 10
  33. 2023Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problemscited by 9
  34. 2024Initialization-enhanced Physics-Informed Neural Network with Domain Decomposition (IDPINN)cited by 9
  35. 2025Neural Tangent Kernel Analysis to Probe Convergence in Physics-informed Neural Solvers: PIKANs vs. PINNscited by 9
  36. 2022Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNscited by 8
  37. 2022Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Mediacited by 8
  38. 2023Transfer learning for improved generalizability in causal physics-informed neural networks for beam simulationscited by 8

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