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A unified deep artificial neural network approach to partial differential equations in complex geometries

arXiv 1711.06464 ↗First posted 17 Nov 2017stat.ML cs.LGcited by 122

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Abstract (arXiv)

In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the network output with respect to the space variables which is needed to approximate the differential operator. The method is based on an ansatz for the solution which requires nothing but feedforward neural networks and an unconstrained gradient based optimization method such as gradient descent or a quasi-Newton method. We show an example where classical mesh based methods cannot be used and neural networks can be seen as an attractive alternative. Finally, we highlight the benefits of deep compared to shallow neural networks and device some other convergence enhancing techniques.

Key facts. Cited by 122 papers in the map (data through 14 Oct 2026). First posted on arXiv 17 Nov 2017.

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Citing papers per year: 2021: 34; 2022: 26; 2023: 21; 2024: 18; 2025: 10; 2026: 13.

122 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  2. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  3. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  4. 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
  5. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  6. 2021Learning in Sinusoidal Spaces with Physics-Informed Neural Networkscited by 62
  7. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 56
  8. 2022Physics Informed Neural Fields for Smoke Reconstruction with Sparse Datacited by 52
  9. 2021Evolutional Deep Neural Networkcited by 50
  10. 2021Physics informed neural networks for continuum micromechanicscited by 46
  11. 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 42
  12. 2023Geometric Clifford Algebra Networkscited by 38
38 more
  1. 2021Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domainscited by 37
  2. 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
  3. 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
  4. 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
  5. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  6. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  7. 2021Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Controlcited by 24
  8. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  9. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
  10. 2021Reproducing Activation Function for Deep Learningcited by 19
  11. 2022Solving Seismic Wave Equations on Variable Velocity Models with Fourier Neural Operatorcited by 18
  12. 2023Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEscited by 18
  13. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  14. 2021Learning optimal multigrid smoothers via neural networkscited by 17
  15. 2023A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networkscited by 16
  16. 2022WarpPINN: Cine-MR image registration with physics-informed neural networkscited by 14
  17. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  18. 2022BI-GreenNet: Learning Green's functions by boundary integral networkcited by 13
  19. 2024Preconditioning for Physics-Informed Neural Networkscited by 13
  20. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  21. 2022A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEscited by 12
  22. 2021Learning Density Distribution of Reachable States for Autonomous Systemscited by 11
  23. 2023BINN: A deep learning approach for computational mechanics problems based on boundary integral equationscited by 11
  24. 2021CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometriescited by 10
  25. 2023Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemescited by 10
  26. 2021A Review of Physics-based Machine Learning in Civil Engineeringcited by 9
  27. 2024Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solverscited by 9
  28. 2021Bayesian neural networks for weak solution of PDEs with uncertainty quantificationcited by 8
  29. 2022CD-ROM: Complemented Deep-Reduced Order Modelcited by 8
  30. 2022Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problemscited by 8
  31. 2022Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Mediacited by 8
  32. 2024A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equationscited by 8
  33. 2021Data-Centric Engineering: integrating simulation, machine learning and statistics. Challenges and Opportunitiescited by 7
  34. 2022An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operatorcited by 7
  35. 2024Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Basescited by 7
  36. 2022Semi-analytic PINN methods for singularly perturbed boundary value problemscited by 6
  37. 2022Neural PDE Solvers for Irregular Domainscited by 6
  38. 2024Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equationscited by 6

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