mlsanityData to 7 Oct 2026

Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features

arXiv 2510.19399 ↗First posted 22 Oct 2025cs.LGcited by 2

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Abstract (arXiv)

Spectral bias, the tendency of neural networks to learn low-frequency features first, is a well-known issue with many training algorithms for physics-informed neural networks (PINNs). To overcome this issue, we propose IFeF-PINN, an algorithm for iterative training of PINNs with Fourier-enhanced features. The key idea is to enrich the latent space using high-frequency components through Random Fourier Features. This creates a two-stage training problem: (i) estimate a basis in the feature space, and (ii) perform regression to determine the coefficients of the enhanced basis functions. For an underlying linear model, it is shown that the latter problem is convex, and we prove that the iterative training scheme converges. Furthermore, we empirically establish that Random Fourier Features enhance the expressive capacity of the network, enabling accurate approximation of high-frequency PDEs. Through extensive numerical evaluation on classical benchmark problems, the superior performance of our method over state-of-the-art algorithms is shown, and the improved approximation across the frequency domain is illustrated.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 27 of its 51 references that are in the map. First posted on arXiv 22 Oct 2025.

Builds on

27 of 51 references.

References · 27

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2018On the Spectral Bias of Neural Networkscited by 1,321
  6. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  7. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  8. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  9. 2015Deep Kernel Learningcited by 440
  10. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  11. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  12. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 142
15 more
  1. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  2. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 66
  3. 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 62
  4. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  5. 2017Linearly constrained Gaussian processescited by 41
  6. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  7. 2022Delta-PINNs: physics-informed neural networks on complex geometriescited by 28
  8. 2024Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspectivecited by 24
  9. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  10. 2023Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEscited by 18
  11. 2024PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representationscited by 11
  12. 2024Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learningcited by 9
  13. 2025A Control Perspective on Training PINNscited by 3
  14. 2025Vanishing Stacked-Residual PINN for State Reconstruction of Hyperbolic Systemscited by 3
  15. 2025RL-PINNs: Reinforcement Learning-Driven Adaptive Sampling for Efficient Training of PINNscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Physics-Informed Neural Networks with Architectural Physics Embedding for Large-Scale Wave Field Reconstructioncited by 1
  2. 2026Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networkscited by 0