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

arXiv: https://arxiv.org/abs/2510.19399 · HTML: https://mlsanity.com/paper/2510.19399/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 51 references that are in the map.
- First posted on arXiv 22 Oct 2025.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [PINNACLE: PINN Adaptive ColLocation and Experimental points selection](https://mlsanity.com/paper/2404.07662/) (2024)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs](https://mlsanity.com/paper/2305.18702/) (2023)
- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learning](https://mlsanity.com/paper/2401.02810/) (2024)
- [A Control Perspective on Training PINNs](https://mlsanity.com/paper/2501.18582/) (2025)
- [Vanishing Stacked-Residual PINN for State Reconstruction of Hyperbolic Systems](https://mlsanity.com/paper/2503.14222/) (2025)
- [RL-PINNs: Reinforcement Learning-Driven Adaptive Sampling for Efficient Training of PINNs](https://mlsanity.com/paper/2504.12949/) (2025)

## Built on by (most cited first, 4 of 4)

- [Physics-Informed Neural Networks with Architectural Physics Embedding for Large-Scale Wave Field Reconstruction](https://mlsanity.com/paper/2603.02231/) (2026)
- [Learning the Stellar Structure Equations via Self-supervised Physics-Informed Neural Networks](https://mlsanity.com/paper/2604.06255/) (2026)
- [Alternating Levenberg-Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features](https://mlsanity.com/paper/2608.05892/) (2026)
- [Operator-informed initialization for Fourier features physics-informed neural networks](https://mlsanity.com/paper/2610.03378/) (2026)

## 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.