Pretraining Basis Functions Accelerates Convergence and Improves Accuracy in Domain-Decomposed Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) provide a flexible approach for solving partial differential equations (PDEs), but suffer from slow training and spectral bias. Domain-decomposed PINNs with random neural network bases can accelerate training by replacing nonlinear optimisation with linear least-squares solves, but their bases can be redundant and poorly suited to a given PDE class. We instead pretrain a shared subdomain basis on example solutions from a PDE class during an offline representation-learning stage, then freeze the basis and solve only for its coefficients at train-time on new problem instances. On 1D harmonic oscillator and 2D Helmholtz problems, pretrained bases significantly improve accuracy, conditioning, and solve efficiency over random bases. Furthermore, we show bases pretrained on low-frequency problems transfer zero-shot to higher-frequency instances by scaling the number of subdomains, without further pretraining.