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

Sean Disarò, Ruma Rani Maity, Aras Bacho

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.

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

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.

Paper Structure

This paper contains 15 sections, 7 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Overview
  4. Methodology
  5. Physics-Informed Machine Learning (PIML)
  6. Hard Constraints For Boundary Conditions
  7. Deflation-PINNs
  8. Numerical Experiments
  9. Landau-De Gennes Model
  10. Reproducibility Statement
  11. Discussion
  12. Acknowledgments
  13. Theorems and Proofs
  14. DeepONets
  15. Figures And Tables

Key Result

Corollary 1

Suppose that $\sigma: \mathbb R\to \mathbb R$ is a continuous and non-polynomial activation function, $X$ is a Banach function space, $u_1, \dots, u_N \in X$ and $K \subset \mathbb R^d$ compact. Then for all $\varepsilon >0$ there exists $p\in \mathbb N$, $\zeta _ i \in \mathbb R$, $\omega_i \in \

Figures (4)

  • Figure 1: Architecture of the Deflation-PINN.
  • Figure 2: Vector field plots of one diagonal and two rotated solutions, denoted by D1, R1 and R3 in suffix, with Deflation PINNs and FEM.
  • Figure 3: Diagonally stable molecular alignments: \ref{['D1']} D1, \ref{['D2']} D2 states and rotated stable molecular alignments: \ref{['R1']} R1, \ref{['R2']} R2, \ref{['R3']} R3, \ref{['R4']} R4 states. See DGFEM for more details.
  • Figure 4: Vector field plots of one diagonal and two rotated solutions, denoted by D1, R1 and R3 in suffix, with Deflation PINNs and FEM.

Theorems & Definitions (15)

  • Definition 1
  • Example 2.1
  • Definition 2
  • Corollary 1
  • proof
  • Theorem 1
  • Corollary 2
  • proof
  • Theorem 2: ChenChenApproxOP
  • Lemma 1
  • ...and 5 more