# Research Program: Theory of Learning in Dynamical Systems

arXiv: https://arxiv.org/abs/2512.19410 · HTML: https://mlsanity.com/paper/2512.19410/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 27 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 10 of 10)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [A Theory of Universal Learning](https://mlsanity.com/paper/2011.04483/) (2020)
- [Learning Linear Dynamical Systems via Spectral Filtering](https://mlsanity.com/paper/1711.00946/) (2017)
- [Online Learning for Time Series Prediction](https://mlsanity.com/paper/1302.6927/) (2013)
- [On the Complexity of Adversarial Decision Making](https://mlsanity.com/paper/2206.13063/) (2022)
- [Universal Sequence Preconditioning](https://mlsanity.com/paper/2502.06545/) (2025)
- [Universal Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2508.11990/) (2025)

## Built on by (most cited first, 5 of 5)

- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Autoregressive Learning in Joint KL: Sharp Oracle Bounds and Lower Bounds](https://mlsanity.com/paper/2605.12316/) (2026)
- [Performative Learning Theory](https://mlsanity.com/paper/2602.04402/) (2026)
- [CLT-Optimal Parameter Error Bounds for Linear System Identification](https://mlsanity.com/paper/2604.21270/) (2026)
- [Quantum-Structured World Models (QSWMs) for Predictive Latent Dynamics](https://mlsanity.com/paper/2608.05371/) (2026)

## Abstract (arXiv)

Modern learning systems increasingly interact with data that evolve over time and depend on hidden internal state. We ask a basic question: when is such a dynamical system learnable from observations alone? This paper proposes a research program for understanding learnability in dynamical systems through the lens of next-token prediction. We argue that learnability in dynamical systems should be studied as a finite-sample question, and be based on the properties of the underlying dynamics rather than the statistical properties of the resulting sequence. To this end, we give a formulation of learnability for stochastic processes induced by dynamical systems, focusing on guarantees that hold uniformly at every time step after a finite burn-in period. This leads to a notion of dynamic learnability which captures how the structure of a system, such as stability, mixing, observability, and spectral properties, governs the number of observations required before reliable prediction becomes possible. We illustrate the framework in the case of linear dynamical systems, showing that accurate prediction can be achieved after finite observation without system identification, by leveraging improper methods based on spectral filtering. We survey the relationship between learning in dynamical systems and classical PAC, online, and universal prediction theories, and suggest directions for studying nonlinear and controlled systems.