Structure Probing Neural Network Deflation

TL;DR

This paper introduces a neural network-based deflation method combined with structure probing to efficiently find multiple solutions of nonlinear differential equations, especially in high-dimensional and complex domains.

Contribution

It proposes a novel structure probing deflation technique that enables neural networks to identify multiple solutions in nonlinear physical models, overcoming the limitation of finding only the flattest local minimizer.

Findings

Capable of solving high-dimensional problems on complex domains.

Finds more solutions than existing methods.

Efficient convergence to multiple solutions.

Abstract

Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This paper proposes a network-based structure probing deflation method to make deep learning capable of identifying multiple solutions that are ubiquitous and important in nonlinear physical models. First, we introduce deflation operators built with known solutions to make known solutions no longer local minimizers of the optimization energy landscape. Second, to facilitate the convergence to the desired local minimizer, a structure probing technique is proposed to obtain an initial guess close to the desired local minimizer. Together with neural network structures carefully designed in this paper, the new regularized optimization can converge to new solutions efficiently. Due to the mesh-free nature of deep learning, the proposed method is capable of solving high-dimensional problems on complicated domains with multiple solutions, while existing methods focus on merely one or two-dimensional regular domains and are more expensive in operation counts. Numerical experiments also demonstrate that the proposed method could find more solutions than exiting methods.