  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.
