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

Physics-informed neural networks often fail on PDEs with high-frequency or multi-scale solutions and strongly nonlinear problems. The authors identify two root causes: spectral bias (underfitting high-frequency features) and representation-coefficient coupling (entanglement of representation learning and coefficient fitting). To counter this, they propose an alternating Levenberg-Marquardt optimization scheme that separates representation learning from coefficient fitting, combined with Fourier-enhanced input features. This approach decouples the optimization steps, reducing the nonconvexity challenges that typically hinder PINN training. The method is validated on benchmark PDE problems, showing improved convergence and solution accuracy compared to standard training. This offers a practical recipe for making PINNs more robust in challenging scientific computing applications.