Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes

TL;DR

This paper introduces Deflation-PINNs, a novel neural network framework that can find multiple solutions to nonlinear PDEs by integrating deflation techniques with PINNs and DeepONets.

Contribution

The paper presents a new method combining deflation loss with PINNs and DeepONets to systematically discover multiple solutions to complex PDEs.

Findings

Successfully identified multiple solutions for the Landau-de Gennes model

Theoretical convergence guarantees for the proposed method

Demonstrated effectiveness in complex energy landscapes

Abstract

Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDE problems, they typically struggle to identify multiple distinct solutions, since they are designed to find one solution at a time. To address this limitation, we introduce Deflation-PINNs, a novel framework that integrates a deflation loss with an architecture based on PINNs and Deep Operator Networks (DeepONets). By incorporating a deflation term into the loss function, our method systematically forces the Deflation-PINN to seek and converge upon distinct finitely many solution branches. We provide theoretical evidence on the convergence of our model and demonstrate the efficacy of Deflation-PINNs through numerical experiments on the Landau-de Gennes model of liquid crystals, a system renowned for its complex energy landscape and multiple equilibrium states. Our results show that Deflation-PINNs can successfully identify and characterize multiple distinct crystal structures.