# Learning dynamical systems from data: Gradient-based dictionary optimization

arXiv: https://arxiv.org/abs/2411.04775 · HTML: https://mlsanity.com/paper/2411.04775/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 7 Nov 2024.

## Builds on (most cited first, 13 of 13)

- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [A Simple Convergence Proof of Adam and Adagrad](https://mlsanity.com/paper/2003.02395/) (2020)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Two methods to approximate the Koopman operator with a reservoir computer](https://mlsanity.com/paper/2008.10263/) (2020)
- [Gradient Descent, Stochastic Optimization, and Other Tales](https://mlsanity.com/paper/2205.00832/) (2022)
- [Towards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithm](https://mlsanity.com/paper/1909.13455/) (2019)
- [Gradient Descent-Type Methods: Background and Simple Unified Convergence Analysis](https://mlsanity.com/paper/2212.09413/) (2022)

## Built on by (most cited first, 3 of 3)

- [Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Engine](https://mlsanity.com/paper/2505.10438/) (2025)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)

## Abstract (arXiv)

The Koopman operator plays a crucial role in analyzing the global behavior of dynamical systems. Existing data-driven methods for approximating the Koopman operator or discovering the governing equations of the underlying system typically require a fixed set of basis functions, also called dictionary. The optimal choice of basis functions is highly problem-dependent and often requires domain knowledge. We present a novel gradient descent-based optimization framework for learning suitable and interpretable basis functions from data and show how it can be used in combination with EDMD, SINDy, and PDE-FIND. We illustrate the efficacy of the proposed approach with the aid of various benchmark problems such as the Ornstein-Uhlenbeck process, Chua's circuit, a nonlinear heat equation, as well as protein-folding data.