An Adaptive Sampling Method to Improve Physics-Informed Neural Networks

Abstract

Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving partial differential equations (PDEs), yet often struggle with localised features due to the random or uniform nature of their training data selection. This thesis introduces a novel adaptive sampling method that reallocates collocation points during training based on a virtual spring model inspired by similar models used for numerical PDE solvers. This spring-inspired reallocation method increases the density of training points in regions with high PDE residuals. Experiments on one-dimensional synthetic benchmarks demonstrate consistent performance gains, especially for problems with steep localised gradients, reducing both training times and solution errors compared to static or random resampling approaches. While the benefits of adaptivity are less significant for smoother problems, the method’s lightweight nature and potential for extension to higher dimensions suggest broad applicability. This work contributes an interpretable and effective tool for enhancing the training of PINNs under computational constraints.