TSPINN: Thompson sampling-based adaptive training for physics-informed neural networks

Physics-Informed Neural Networks (PINNs) offer a powerful framework for solving partial differential equations (PDEs) but often suffer from training inefficiencies. A critical factor is the selection of effective sampling distributions for training points. This work introduces Thompson Sampling PINN (TSPINN), a novel adaptive sampling approach inspired by Thompson Sampling. TSPINN dynamically defines a reward distribution based on recent PDE residuals to guide the adaptive selection of training points, strategically balancing exploration of uncertain regions and exploitation of high-residual zones. Furthermore, we integrate TSPINN with the Causal PINN framework to develop Causal TSPINN, explicitly incorporating temporal and spatial causality into the sampling process. Extensive numerical experiments demonstrate that both TSPINN and Causal TSPINN significantly enhance solution accuracy while requiring fewer training iterations and collocation points compared to existing adaptive sampling techniques (e.g., R3, RAD). These methods demonstrate broad applicability across various PDE types, including Poisson, Helmholtz, and convection equations, among others.