New PINN Training Boosts High-Frequency PDE Solutions

Physics-informed neural networks (PINNs) are powerful tools for solving partial differential equations (PDEs), but they often struggle with problems involving high-frequency, multi-scale, or strongly nonlinear solutions. This difficulty stems from two main issues: spectral bias, where neural networks tend to underfit high-frequency features, and representation-coefficient coupling, where learning the data representation and fitting coefficients are entangled in a single, complex optimization problem. Researchers have proposed the Fourier-enhanced alternating Levenberg-Marquardt PINN (FALM-PINN) to overcome these limitations. This novel optimization framework effectively decouples the representation learning process from the coefficient fitting. The "upper-level" problem focuses on learning a Fourier-enhanced basis, which enriches the latent space with the necessary high-frequency components. Concurrently, the "lower-level" problem resolves the coupling by fitting the projection coefficients on this enhanced basis, utilizing the Levenberg-Marquardt algorithm to solve a nonlinear least-squares problem. This framework is applicable to general nonlinear and coupled PDE systems and simplifies to a single-step convex optimization for linear PDEs. The study provides a proof of global convergence for this alternating training scheme in both scenarios. Numerical experiments on challenging high-frequency and nonlinear PDEs demonstrate that FALM-PINN achieves relative L2 errors up to two orders of magnitude lower than state-of-the-art baselines, marking a significant advancement in PINN accuracy.