Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes
This addresses a limitation in PINNs for researchers in mathematical physics and engineering dealing with systems like liquid crystals that have multiple equilibrium states, representing a novel method for a known bottleneck.
The paper tackles the problem of Physics-Informed Neural Networks (PINNs) struggling to identify multiple distinct solutions for nonlinear PDEs by introducing Deflation-PINNs, which integrate a deflation loss with PINNs and DeepONets to systematically find multiple solution branches, as demonstrated on the Landau-de Gennes model of liquid crystals.
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.