mlsanityData to 14 Oct 2026

Universal Learning of Nonlinear Dynamics

arXiv 2508.11990 ↗First posted 16 Aug 2025cs.LG math.OC stat.MLcited by 5

Not on the map. Why?

Abstract (arXiv)

We study the fundamental problem of learning a marginally stable unknown nonlinear dynamical system. We describe an algorithm for this problem, based on the technique of spectral filtering, which learns a mapping from past observations to the next based on a spectral representation of the system. Using techniques from online convex optimization, we prove vanishing prediction error for any nonlinear dynamical system that has finitely many marginally stable modes, with rates governed by a novel quantitative control-theoretic notion of learnability. The main technical component of our method is a new spectral filtering algorithm for linear dynamical systems, which incorporates past observations and applies to general noisy and marginally stable systems. This significantly generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 27 of its 50 references that are in the map. First posted on arXiv 16 Aug 2025.

Builds on

27 of 50 references.

References · 27

  1. 2017Attention Is All You Needcited by 60,361
  2. 2023Mamba: Linear-Time Sequence Modeling with Selective State Spacescited by 4,718
  3. 2019Introduction to Online Convex Optimizationcited by 803
  4. 2021Brax -- A Differentiable Physics Engine for Large Scale Rigid Body Simulationcited by 407
  5. 2023Hyena Hierarchy: Towards Larger Convolutional Language Modelscited by 393
  6. 2018Learning Without Mixing: Towards A Sharp Analysis of Linear System Identificationcited by 158
  7. 2020Naive Exploration is Optimal for Online LQRcited by 116
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  9. 2016Gradient Descent Learns Linear Dynamical Systemscited by 79
  10. 2018Non-asymptotic Identification of LTI Systems from a Single Trajectorycited by 78
  11. 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 67
  12. 2013Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributionscited by 60
15 more
  1. 2019Learning Linear Dynamical Systems with Semi-Parametric Least Squarescited by 58
  2. 2019The Nonstochastic Control Problemcited by 52
  3. 2017Learning Linear Dynamical Systems via Spectral Filteringcited by 46
  4. 2020Black-Box Control for Linear Dynamical Systemscited by 43
  5. 2018Spectral Filtering for General Linear Dynamical Systemscited by 37
  6. 2013Online Learning for Time Series Predictioncited by 33
  7. 2023A New Approach to Learning Linear Dynamical Systemscited by 23
  8. 2023Spectral State Space Modelscited by 21
  9. 2020No-Regret Prediction in Marginally Stable Systemscited by 18
  10. 2018On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filterscited by 16
  11. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  12. 2023Tensor Decompositions Meet Control Theory: Learning General Mixtures of Linear Dynamical Systemscited by 10
  13. 2018Theory and Algorithms for Forecasting Time Seriescited by 7
  14. 2024Flash STU: Fast Spectral Transform Unitscited by 7
  15. 2020Luenberger observers for discrete-time nonlinear systemscited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 4.

5 papers.

  1. 2025Research Program: Theory of Learning in Dynamical Systemscited by 5
  2. 2026The Power of Second Order Methods for Sequence Preconditioningcited by 1
  3. 2026SFO: Learning PDE Operators via Spectral Filteringcited by 0
  4. 2026Two-Layer Linear Auto-Regressive Models Estimate Latent Statescited by 0
  5. 2026Spectral Distillation: From Nonlinear Dynamics to Linear State-Space Modelscited by 0