mlsanityData to 14 Oct 2026

Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Bases

arXiv 2410.03496 ↗First posted 4 Oct 2024cs.LGcited by 7

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Abstract (arXiv)

Interest is rising in Physics-Informed Neural Networks (PINNs) as a mesh-free alternative to traditional numerical solvers for partial differential equations (PDEs). However, PINNs often struggle to learn high-frequency and multi-scale target solutions. To tackle this problem, we first study a strong Boundary Condition (BC) version of PINNs for Dirichlet BCs and observe a consistent decline in relative error compared to the standard PINNs. We then perform a theoretical analysis based on the Fourier transform and convolution theorem. We find that strong BC PINNs can better learn the amplitudes of high-frequency components of the target solutions. However, constructing the architecture for strong BC PINNs is difficult for many BCs and domain geometries. Enlightened by our theoretical analysis, we propose Fourier PINNs -- a simple, general, yet powerful method that augments PINNs with pre-specified, dense Fourier bases. Our proposed architecture likewise learns high-frequency components better but places no restrictions on the particular BCs or problem domains. We develop an adaptive learning and basis selection algorithm via alternating neural net basis optimization, Fourier and neural net basis coefficient estimation, and coefficient truncation. This scheme can flexibly identify the significant frequencies while weakening the nominal frequencies to better capture the target solution's power spectrum. We show the advantage of our approach through a set of systematic experiments.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 32 of its 67 references that are in the map. First posted on arXiv 4 Oct 2024.

Builds on

32 of 67 references.

References · 32

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  4. 2018On the Spectral Bias of Neural Networkscited by 1,341
  5. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  6. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  7. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  8. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  9. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  10. 2019Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networkscited by 368
  11. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  12. 2021Physics-informed neural networks with hard constraints for inverse designcited by 280
20 more
  1. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  2. 2015Tensorizing Neural Networkscited by 250
  3. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 233
  4. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
  5. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  6. 2020Frequency Bias in Neural Networks for Input of Non-Uniform Densitycited by 140
  7. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
  8. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 135
  9. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 129
  10. 2020Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networkscited by 123
  11. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  12. 2020Neural Pruning via Growing Regularizationcited by 97
  13. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 72
  14. 2020A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecturecited by 50
  15. 2018On the Convergence of the SINDy Algorithmcited by 49
  16. 2019Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpointcited by 42
  17. 2023Domain Agnostic Fourier Neural Operatorscited by 38
  18. 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
  19. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  20. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 6.

7 papers.

  1. 2025AMR-Transformer: Enabling Efficient Long-range Interaction for Complex Neural Fluid Simulationcited by 8
  2. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  3. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  4. 2026High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equationcited by 0
  5. 2026Pseudo-differential-enhanced physics-informed neural networkscited by 0
  6. 2026Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEscited by 0
  7. 2026A Novel Fourier Feature Network for Solving Partial Differential Equationscited by 0