# Long-time predictive modeling of nonlinear dynamical systems using neural networks

arXiv: https://arxiv.org/abs/1805.12547 · HTML: https://mlsanity.com/paper/1805.12547/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2018.

## Built on by (most cited first, 15 of 15)

- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Model Order Reduction based on Runge-Kutta Neural Network](https://mlsanity.com/paper/2103.13805/) (2021)
- [On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver](https://mlsanity.com/paper/2208.03680/) (2022)
- [Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators](https://mlsanity.com/paper/2106.09658/) (2021)
- [Initializing LSTM internal states via manifold learning](https://mlsanity.com/paper/2104.13101/) (2021)
- [Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint](https://mlsanity.com/paper/2109.11446/) (2021)
- [Comparison of neural closure models for discretised PDEs](https://mlsanity.com/paper/2210.14675/) (2022)
- [Neural ODEs with Irregular and Noisy Data](https://mlsanity.com/paper/2205.09479/) (2022)
- [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)
- [Active-learning-based non-intrusive Model Order Reduction](https://mlsanity.com/paper/2204.08523/) (2022)
- [Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solvers](https://mlsanity.com/paper/2106.02543/) (2021)
- [A Novel Neural Filter to Improve Accuracy of Neural Network Models of Dynamic Systems](https://mlsanity.com/paper/2409.13654/) (2024)
- [Fast Deep Predictive Coding Networks for Videos Feature Extraction without Labels](https://mlsanity.com/paper/2409.04945/) (2024)
- [Latent-Lagrangian Neural Networks for Reduced Order Modeling of Non-autonomous Nonlinear Dynamical Systems](https://mlsanity.com/paper/2610.04723/) (2026)

## Abstract (arXiv)

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of the strong correlation between the local error and the maximum singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder, and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent.