DONIS: Importance Sampling for Training Physics-Informed DeepONet

Abstract

Deep Operator Network (DeepONet) effectively learns complex operator mappings, especially for systems governed by differential equations. Physics-informed DeepONet (PI-DeepONet) extends these capabilities by integrating physical constraints, enabling robust performance with limited or no labeled data. However, combining operator learning with these constraints increases computational complexity, which makes training more difficult and convergence slower, particularly for nonlinear or high-dimensional problems. In this work, we present an enhanced PI-DeepONet framework, that applies importance sampling to both of DeepONet inputs (i.e., the functions and the collocation points) to alleviate these training challenges. By focusing on critical data regions in both input domains, our approach showcases accelerated convergence and improved accuracy across various complex applications.