# Model Order Reduction based on Runge-Kutta Neural Network

arXiv: https://arxiv.org/abs/2103.13805 · HTML: https://mlsanity.com/paper/2103.13805/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 46 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 8 of 8)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Digital Twin: Values, Challenges and Enablers](https://mlsanity.com/paper/1910.01719/) (2019)
- [DR-RNN: A deep residual recurrent neural network for model reduction](https://mlsanity.com/paper/1709.00939/) (2017)
- [Long-time predictive modeling of nonlinear dynamical systems using neural networks](https://mlsanity.com/paper/1805.12547/) (2018)

## Built on by (most cited first, 9 of 9)

- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)
- [A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems](https://mlsanity.com/paper/2208.04995/) (2022)
- [TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems](https://mlsanity.com/paper/2412.07010/) (2024)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [Real-time Human Response Prediction Using a Non-intrusive Data-driven Model Reduction Scheme](https://mlsanity.com/paper/2110.13583/) (2021)
- [Active-learning-based non-intrusive Model Order Reduction](https://mlsanity.com/paper/2204.08523/) (2022)
- [Recurrent Deep Kernel Learning of Dynamical Systems](https://mlsanity.com/paper/2405.19785/) (2024)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)

## Abstract (arXiv)

Model Order Reduction (MOR) methods enable the generation of real-time-capable digital twins, which can enable various novel value streams in industry. While traditional projection-based methods are robust and accurate for linear problems, incorporating Machine Learning to deal with nonlinearity becomes a new choice for reducing complex problems. Such methods usually consist of two steps. The first step is dimension reduction by projection-based method, and the second is the model reconstruction by Neural Network. In this work, we apply some modifications for both steps respectively and investigate how they are impacted by testing with three simulation models. In all cases Proper Orthogonal Decomposition (POD) is used for dimension reduction. For this step, the effects of generating the input snapshot database with constant input parameters is compared with time-dependent input parameters. For the model reconstruction step, two types of neural network architectures are compared: Multilayer Perceptron (MLP) and Runge-Kutta Neural Network (RKNN). The MLP learns the system state directly while RKNN learns the derivative of system state and predicts the new state as a Runge-Kutta integrator.