Newton-PINet: A fast physics-informed neural network with Newton linearization for meta-learning nonlinear PDEs

Scientific machine learning has opened new avenues for solving parameterized partial differential equations (PDEs), enabling models to learn a family of PDEs and generalize to unseen instances. In this context, data-driven operator learning methods typically require large training datasets, while physics-informed neural networks (PINNs) suffer from difficult optimization and limited generalization, especially for nonlinear PDEs. We propose Newton-PINet, a physics-informed network enhanced by Newton linearization, offering an effective meta-learning framework for nonlinear PDEs. Newton-PINet (i) employs a physics-informed multilayer network with skip connections, where the output-layer weights are solved by least squares; (ii) adopts a two-stage learning strategy that first leverages gradient-based training to learn robust representations from the available training tasks, and then performs gradient-free fine-tuning on the output layer for fast task-specific generalization; and (iii) incorporates a Newton linearization method to speed up the least-squares iteration for nonlinear PDE problems. On a challenging nonlinear reaction-diffusion benchmark, Newton-PINet achieves up to three orders of magnitude lower relative error than recent neural solvers, while using 16× fewer training tasks and over an order of magnitude less training time (under 5 minutes versus several hours). This work advances the meta-learning of PINNs toward data-efficient, fast, and generalizable physics solvers. The datasets and code are provided in the supplementary material.