Alternating Levenberg-Marquardt Training of Physics-Informed Neural Networks with Fourier-Enhanced Features

2026-08-06

AI Summary

This work addresses the limited accuracy of physics-informed neural networks (PINNs) when solving high-frequency, multiscale, or strongly nonlinear partial differential equations (PDEs), which stems from spectral bias and the coupling between representation learning and coefficient fitting. To overcome these challenges, we propose the FALM-PINN framework, which decouples representation learning from coefficient optimization for the first time. In the upper layer, Fourier feature augmentation constructs basis functions enriched with high-frequency components to mitigate spectral bias; in the lower layer, the Levenberg–Marquardt algorithm efficiently solves a nonlinear least-squares problem over this enhanced basis. The resulting bilevel alternating optimization scheme guarantees global convergence and is applicable to both linear and nonlinear PDEs. Numerical experiments demonstrate that FALM-PINN achieves up to two orders of magnitude lower relative L² error compared to state-of-the-art methods across multiple benchmark problems.

Abstract

Physics-informed neural networks (PINNs) often fail to accurately resolve partial differential equations (PDEs) with high-frequency or multi-scale solutions, as well as strongly nonlinear problems. Two factors underlie this difficulty: spectral bias, the tendency of neural networks to underfit high-frequency features; and representation-coefficient coupling, the entanglement of representation learning and coefficient fitting within a single nonconvex optimization objective. In this work, we propose the Fourier-enhanced alternating Levenberg--Marquardt PINN (FALM-PINN), an optimization framework that decouples representation learning from coefficient fitting. The upper-level problem learns a Fourier-enhanced basis that enriches the latent space with high-frequency components, while the lower-level problem resolves the coupling by fitting the projection coefficients on this basis, solving a nonlinear least-squares problem with the Levenberg--Marquardt algorithm. The framework applies to general nonlinear and coupled PDE systems, and reduces to a single-step convex optimization problem for linear PDEs. We prove global convergence of the alternating training scheme in both cases. Numerical examples on multiple challenging high-frequency and nonlinear PDEs show that FALM-PINN achieves relative $L^2$ errors up to two orders of magnitude lower than state-of-the-art baselines.

Problem

• Physics-informed neural networks
• high-frequency solutions
• spectral bias
• representation-coefficient coupling
• nonlinear PDEs

Innovation

• Fourier-enhanced features
• alternating optimization
• Levenberg-Marquardt algorithm
• spectral bias mitigation
• physics-informed neural networks