Neural empirical interpolation method for nonlinear model reduction

TL;DR

The paper introduces NEIM, a neural network-based method for efficient nonlinear model reduction in parameterized PDEs, offering a data-driven, easy-to-implement alternative to traditional methods with demonstrated effectiveness.

Contribution

NEIM is a novel neural network-based greedy algorithm that approximates nonlinear terms in reduced order models, improving efficiency and ease of implementation.

Findings

Effective on various nonlinear PDEs

Reduces computational complexity of nonlinear terms

Compatible with automatic differentiation tools

Abstract

In this paper, we introduce the neural empirical interpolation method (NEIM), a neural network-based alternative to the discrete empirical interpolation method for reducing the time complexity of computing the nonlinear term in a reduced order model (ROM) for a parameterized nonlinear partial differential equation. NEIM is a greedy algorithm which accomplishes this reduction by approximating an affine decomposition of the nonlinear term of the ROM, where the vector terms of the expansion are given by neural networks depending on the ROM solution, and the coefficients are given by an interpolation of some "optimal" coefficients. Because NEIM is based on a greedy strategy, we are able to provide a basic error analysis to investigate its performance. NEIM has the advantages of being easy to implement in models with automatic differentiation, of being a nonlinear projection of the ROM nonlinearity, of being efficient for both nonlocal and local nonlinearities, and of relying solely on data and not the explicit form of the ROM nonlinearity. We demonstrate the effectiveness of the methodology on solution-dependent and solution-independent nonlinearities, a nonlinear elliptic problem, and a nonlinear parabolic model of liquid crystals. Code availability: https://github.com/maxhirsch/NEIM