mlsanityData to 7 Oct 2026

Learning dynamical systems from data: Gradient-based dictionary optimization

arXiv 2411.04775 ↗First posted 7 Nov 2024math.DS cs.LG stat.MLcited by 3

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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.

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

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13 of 38 references.

References · 13

  1. 2018On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimizationcited by 173
  2. 2020A Simple Convergence Proof of Adam and Adagradcited by 169
  3. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  4. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  5. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  6. 2017Sparse learning of stochastic dynamic equationscited by 76
  7. 2017Variational approach for learning Markov processes from time series datacited by 51
  8. 2017Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spacescited by 48
  9. 2016Linear identification of nonlinear systems: A lifting technique based on the Koopman operatorcited by 18
  10. 2020Two methods to approximate the Koopman operator with a reservoir computercited by 7
  11. 2022Gradient Descent, Stochastic Optimization, and Other Talescited by 3
  12. 2019Towards Scalable Koopman Operator Learning: Convergence Rates and A Distributed Learning Algorithmcited by 1
1 more
  1. 2022Gradient Descent-Type Methods: Background and Simple Unified Convergence Analysiscited by 1

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Citing papers per year: 2025: 3.

3 papers.

  1. 2025Koopman Eigenfunction-Based Identification and Optimal Nonlinear Control of Turbojet Enginecited by 2
  2. 2025How deep is your network? Deep vs. shallow learning of transfer operatorscited by 2
  3. 2025Bayesian Transfer Operators in Reproducing Kernel Hilbert Spacescited by 0