# Neural empirical interpolation method for nonlinear model reduction

arXiv: https://arxiv.org/abs/2406.03562 · HTML: https://mlsanity.com/paper/2406.03562/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 5 Jun 2024.

## Builds on (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Stop using the elbow criterion for k-means and how to choose the number of clusters instead](https://mlsanity.com/paper/2212.12189/) (2022)
- [An artificial neural network approach to bifurcating phenomena in computational fluid dynamics](https://mlsanity.com/paper/2109.10765/) (2021)
- [Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs](https://mlsanity.com/paper/2202.02658/) (2022)
- [A Note on Dimensionality Reduction in Deep Neural Networks using Empirical Interpolation Method](https://mlsanity.com/paper/2305.09842/) (2023)

## Built on by (most cited first, 4 of 4)

- [Optimal Transport-Based Displacement Interpolation with Data Augmentation for Reduced Order Modeling of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2411.08750/) (2024)
- [Convergence and Sketching-Based Efficient Computation of Neural Tangent Kernel Weights in Physics-Based Loss](https://mlsanity.com/paper/2511.15530/) (2025)
- [Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition](https://mlsanity.com/paper/2511.23037/) (2025)
- [Integrating Artificial Intelligence, Physics, and Internet of Things: A Framework for Cultural Heritage Conservation](https://mlsanity.com/paper/2604.03233/) (2026)

## Abstract (arXiv)

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