🤖 AI Summary
This work addresses the high computational cost of evaluating nonlinear terms in reduced-order models (ROMs) for parametrized nonlinear partial differential equations (PDEs). We propose the Neural Empirical Interpolation Method (NEIM), a data-driven framework that employs neural networks to automatically construct affine decompositions of nonlinear operators—without requiring explicit functional forms—and uniformly handles both solution-dependent and solution-independent nonlinearities, including local and nonlocal cases. Its key innovation is a greedy algorithm-driven neural empirical interpolation framework, fully compatible with automatic differentiation, which drastically reduces online computational complexity. Numerical experiments on a nonlinear elliptic equation and a parabolic model for liquid crystal phase transitions demonstrate that NEIM achieves substantial compression of nonlinear term evaluation cost while preserving high accuracy. The implementation is publicly available.
📝 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