# Learning Dynamical Systems with Side Information

arXiv: https://arxiv.org/abs/2008.10135 · HTML: https://mlsanity.com/paper/2008.10135/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Aug 2020.

## Built on by (most cited first, 19 of 19)

- [SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning](https://mlsanity.com/paper/2403.09110/) (2024)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Machine learning for industrial sensing and control: A survey and practical perspective](https://mlsanity.com/paper/2401.13836/) (2024)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Learning to Predict 3D Rotational Dynamics from Images of a Rigid Body with Unknown Mass Distribution](https://mlsanity.com/paper/2308.14666/) (2023)
- [Learning Interpretable Dynamics from Images of a Freely Rotating 3D Rigid Body](https://mlsanity.com/paper/2209.11355/) (2022)
- [Stable Inverse Reinforcement Learning: Policies from Control Lyapunov Landscapes](https://mlsanity.com/paper/2405.08756/) (2024)
- [Approximation of optimization problems with constraints through kernel Sum-Of-Squares](https://mlsanity.com/paper/2301.06339/) (2023)
- [Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization](https://mlsanity.com/paper/2303.01483/) (2023)
- [Learning of Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2312.09734/) (2023)
- [Learning Hamiltonian Dynamics with Reproducing Kernel Hilbert Spaces and Random Features](https://mlsanity.com/paper/2404.07703/) (2024)
- [Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data](https://mlsanity.com/paper/2512.23761/) (2025)
- [Learning to Reach, Swim, Walk and Fly in One Trial: Data-Driven Control with Scarce Data and Side Information](https://mlsanity.com/paper/2106.10533/) (2021)
- [Physics-Informed Kernel Embeddings: Integrating Prior System Knowledge with Data-Driven Control](https://mlsanity.com/paper/2301.03565/) (2023)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)
- [Learning dynamical systems with biochemically informed neural ordinary differential equations](https://mlsanity.com/paper/2605.24170/) (2026)
- [Safely Learning Dynamical Systems](https://mlsanity.com/paper/2305.12284/) (2023)
- [Data-Scarce Identification of Game Dynamics via Sum-of-Squares Optimization](https://mlsanity.com/paper/2307.06640/) (2023)
- [Learning Polynomial Representations of Physical Objects with Application to Certifying Correct Packing Configurations](https://mlsanity.com/paper/2312.06791/) (2023)

## Abstract (arXiv)

We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or basic principles of a scientific discipline. We are interested in explicitly integrating side information into the learning process in order to compensate for scarcity of trajectory observations. We identify six types of side information that arise naturally in many applications and lead to convex constraints in the learning problem. First, we show that when our model for the unknown dynamical system is parameterized as a polynomial, one can impose our side information constraints computationally via semidefinite programming. We then demonstrate the added value of side information for learning the dynamics of basic models in physics and cell biology, as well as for learning and controlling the dynamics of a model in epidemiology. Finally, we study how well polynomial dynamical systems can approximate continuously-differentiable ones while satisfying side information (either exactly or approximately). Our overall learning methodology combines ideas from convex optimization, real algebra, dynamical systems, and functional approximation theory, and can potentially lead to new synergies between these areas.