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

Yuchen Fan, Chang Wei, Pao-Hsiung Chiu, Chin Chun Ooi, Heyang Wang, Jian Cheng Wong

Keywords: Physics-informed neural networks; nonlinear PDEs; data-efficient meta-learning; fast generalization

TL;DR: In meta-learning nonlinear PDEs, Newton-PINet achieves high generalization accuracy while requiring fewer training costs than state-of-the-art baselines.

Abstract: 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 data, while physics-informed neural networks (PINNs) trained with PDE-based loss functions suffer from challenging optimization landscapes and limited generalization, especially for nonlinear PDEs. To resolve these issues, we develop Newton-PINet, a physics-informed network enhanced by Newton linearization, offering an effective meta-learning framework for nonlinear PDEs. It (i) introduces a physics-informed multilayer network with skip connections from early hidden layers to the output, where the final-layer weights are computed using least-squares method; (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. Newton-PINet achieves relative errors three orders of magnitude lower than recent neural solver baselines on a challenging nonlinear reaction-diffusion benchmark, even while using 16$\times$ fewer training tasks and an order of magnitude less training time (under 2 minutes against the several hours these baselines required). This work advances the meta-learning of PINNs toward data-efficient, fast, and generalizable physics solvers.

Primary Area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

Submission Number: 9579