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.

我们介绍了神经经验插值方法（NEIM），这是一种基于神经网络的替代离散经验插值方法，用于降低计算参数化非线性偏微分方程的约减阶模型（ROM）中的非线性项的时间复杂度。 NEIM是一种贪婪算法，通过逼近ROM的非线性项的仿射分解来实现约减，其中展开的向量项由取决于ROM解的神经网络给出，系数由某些“最优”系数的插值给出。由于NEIM基于贪婪策略，我们能够提供基本的误差分析以研究其性能。 NEIM具有易于在具有自动微分的模型中实现、是ROM非线性的非线性投影、对非局部和局部非线性都高效以及仅依赖数据而不依赖ROM非线性的明确形式的优点。我们在依赖解和不依赖解的非线性、非线性椭圆问题以及液晶的非线性抛物模型上展示了该方法的有效性。

非线性模型简化的神经经验插值方法