Title: Pareto-Guided Regional Sampling for Adaptive Collocation in Physics-Informed Neural Networks
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: Physics-informed neural networks; Adaptive sampling; Collocation points

Abstract: Physics-informed neural networks (PINNs) provide a powerful framework for solving both forward and inverse problems of differential equations by embedding physical constraints in the loss function. However, in regions that exhibit localized stiffness or sharp gradients, conventional PINN training using uniformly sampled collocation points often leads to suboptimal training efficiency and predictive accuracy. To address this challenge, this work proposes a Pareto-guided regional sampling (PaRS) framework for adaptive collocation point sampling and tuning. The proposed method integrates three key components: residual decomposition, adaptive loss weighting, and dynamic resampling. Specifically, the spatial domain of collocation points is first partitioned into multiple subregions, and the training objectives, such as residuals from different subregions, are treated as competing objectives. As a result, a Pareto front is constructed to capture trade-offs among competing residual and other losses. Subsequently, a novel Pareto-guided weighting method is developed to assign adaptive weights to each objective, informed by training progress and information across the Pareto front. These weights further guide the resampling of collocation points in each region, which increases the sampling density in underexplored or error-prone regions. The results of the experiment demonstrate that the proposed PaRS-PINN performs better than standard PINNs and state-of-the-art adaptive collocation point methods.

TL;DR: A Pareto-guided regional sampling (PaRS) framework is proposed for adaptive collocation point sampling and tuning.