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.

翻译：本文介绍了神经经验插值方法（NEIM），这是一种基于神经网络的离散经验插值方法替代方案，用于降低参数化非线性偏微分方程降阶模型（ROM）中非线性项计算的时间复杂度。NEIM采用贪心算法，通过逼近ROM非线性项的仿射分解来实现降阶，其中展开的向量项由依赖于ROM解的神经网络给出，系数则通过对某些“最优”系数的插值获得。由于NEIM基于贪心策略，我们能够提供基础误差分析以评估其性能。该方法具有以下优势：易于在支持自动微分的模型中实现；构成ROM非线性项的非线性投影；对非局部和局部非线性均具有高效性；仅依赖数据而不需要ROM非线性项的显式形式。我们通过解依赖型与解独立型非线性问题、非线性椭圆问题以及液晶的非线性抛物模型验证了该方法的有效性。