# Neural Dynamical Systems: Balancing Structure and Flexibility in Physical Prediction

arXiv: https://arxiv.org/abs/2006.12682 · HTML: https://mlsanity.com/paper/2006.12682/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jun 2020.

## Built on by (most cited first, 18 of 18)

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- [Learning Reduced Nonlinear State-Space Models: an Output-Error Based Canonical Approach](https://mlsanity.com/paper/2206.04791/) (2022)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Hybrid Generative Modeling for Incomplete Physics: Deep Grey-Box Meets Optimal Transport](https://mlsanity.com/paper/2506.22204/) (2025)
- [Knowledge-Informed Kernel State Reconstruction from Heterogeneous Partial Observations](https://mlsanity.com/paper/2601.22328/) (2026)
- [Sharpness-Aware Hybrid Model Learning for Architecture-Agnostic Parameter Estimation](https://mlsanity.com/paper/2602.06837/) (2026)
- [Climate Physics Dynamic Matching](https://mlsanity.com/paper/2608.26907/) (2026)

## Abstract (arXiv)

We introduce Neural Dynamical Systems (NDS), a method of learning dynamical models in various gray-box settings which incorporates prior knowledge in the form of systems of ordinary differential equations. NDS uses neural networks to estimate free parameters of the system, predicts residual terms, and numerically integrates over time to predict future states. A key insight is that many real dynamical systems of interest are hard to model because the dynamics may vary across rollouts. We mitigate this problem by taking a trajectory of prior states as the input to NDS and train it to dynamically estimate system parameters using the preceding trajectory. We find that NDS learns dynamics with higher accuracy and fewer samples than a variety of deep learning methods that do not incorporate the prior knowledge and methods from the system identification literature which do. We demonstrate these advantages first on synthetic dynamical systems and then on real data captured from deuterium shots from a nuclear fusion reactor. Finally, we demonstrate that these benefits can be utilized for control in small-scale experiments.