Solving large-scale variational inequalities with dynamically adjusting initial condition in physics-informed neural networks

Abstract

This work aims to solve large-scale variational inequalities (VIs), which are equivalent to high-dimensional systems of ordinary differential equations (ODEs). The existing physics-informed neural network (PINN) approach (Wu and Lisser, 2023) shows superior performance for VIs with less than 1000 variables, but fails for VIs of larger size, due to the increasing number of equations and the requirement of an extensive time interval. To overcome this limitation, we present two algorithms that dynamically adjust the initial condition for the PINN. The first algorithm uses multiple PINNs sequentially to decompose the task, where the best prediction from the current PINN serves as the initial condition for the next PINN. The second algorithm uses a single PINN throughout the solution process, immediately taking any improved prediction as an initial condition and refining the PINN to achieve a better prediction. Finally, we demonstrate the effectiveness of the proposed algorithms on a number of large-scale VI problems with up to 100,000 variables.