Universal Learning of Nonlinear Dynamics

Evan Dogariu; Anand Brahmbhatt; Elad Hazan

cs.LGAug 16, 2025

Universal Learning of Nonlinear Dynamics

Authors: Evan Dogariu, Anand Brahmbhatt, Elad Hazan

Organizations: Courant Institute of Mathematical Sciences, New York University · Princeton University

Abstract

We study the fundamental problem of one-step prediction of 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 nonexpansive nonlinear dynamical system with 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 generalizes the original spectral filtering algorithm to both asymmetric dynamics as well as incorporating noise correction, and is of independent interest.

Figures & tables

	$\displaystyle\hat{y}_{t}(\Gamma^{t})$	$\displaystyle=\sum_{j=1}^{m-1}J_{j}^{t}u_{t-j}+\sum_{i=1}^{h}\sigma_{i}^{1/4}M_{i}^{t}\langle\phi_{i},u_{t-2:t-T}\rangle$
		$\displaystyle\;+\sum_{j=1}^{m}P_{j}^{t}y_{t-j}+\sum_{i=1}^{h}\sigma_{i}^{1/4}N_{i}^{t}\langle\phi_{j},y_{t-1:t-T-1}\rangle\,,$

Algorithm 1 Observation Spectral Filtering + Regression

Panel of Figure 1 — Figure 1: Instantaneous losses $\left\lVert\hat{y}_{t}-y_{t}\right\rVert^{2}$ plotted on log scale as a function of $t$ for the fully and partially observed Lorenz system, respectively, averaged over 12 random seeds with a smoothing filter of length 1000. Shading indicates $\pm 1$ standard deviation.

Panel of Figure 2 — Figure 2: Two sets of autoregressive trajectories of length 512, plotted alongside the ground truth trajectory given by continuing to simulate the Lorenz ODE. The initial positions at which the rollouts start are marked with a red star.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Marginally stable	$d_{\text{hidden}}$ -free	asymmetric	adversarial noise
Sys-ID	$\times$	$\times$	$\checkmark$	$\times$
Regression, open-loop	$\times$	$\checkmark$	$\checkmark$	$\checkmark$
Regression, closed-loop	$\checkmark$	$\times$	$\checkmark$	$\checkmark$
Spectral filtering ( Hazan et al., 2017 )	$\checkmark$	$\checkmark$	$\times$	$\times$
USP ( Marsden and Hazan, 2025 )	$\checkmark$	$\checkmark$	$\sim\checkmark$	$\times$
OSF (ours)	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$

Appendix

Table 1: Methods for learning LDS.

Panel of Figure 3 — Figure 3: Instantaneous losses $\|\hat{y}_{t}-y_{t}\|^{2}$ plotted as a function of $t$ for experiments (1) and (2), averaged over 12 random seeds with a smoothing filter of length 100 and 1000, respectively. Shading indicates $\pm 1$ standard deviation.

Panel of Figure 4 — Figure 4: Instantaneous losses $\left\lVert\hat{y}_{t}-y_{t}\right\rVert^{2}$ plotted on log scale as a function of $t$ for the fully and partially observed double pendulum, respectively, averaged over 12 random seeds with a smoothing filter of length 1000. Shading indicates $\pm 1$ standard deviation.

Panel of Figure 5 — Figure 5: Predictions $\hat{y}_{t}$ of the four algorithms along with the ground truth $y_{t}$ plotted as a function of $t$ for the fully and partially observed double pendulum, respectively, on a single random seed. We zoom in on the last 100 steps of the trajectory. For the full observation case, since we cannot plot the full state we choose to plot the sine of the angle between the two arms of the pendulum, which is a representative coordinate for the task of next-state prediction.

Panel of Figure 6 — Figure 6: Densities of the stationary distribution $\pi$ corresponding to the chosen potential $V$ for $d_{\mathcal{X}}$ equal to 1 and 2, respectively. We see a number of asymmetric wells that is exponential in the dimension.

Figure 7: Instantaneous losses $\left\lVert\hat{y}_{t}-y_{t}\right\rVert^{2}$ plotted on log scale as a function of $t$ for the Langevin dynamics, averaged over 12 random seeds with a smoothing filter of length 200. Shading indicates $\pm 1$ standard deviation.

Panel of Figure 8 — Figure 8: Eigenvalues of the lifted linear dynamics learned by the eDMD algorithm on the Lorenz system, double pendulum, and Langevin dynamics, respectively. The $x$ and $y$ axes display real and complex components, respectively, and we draw the unit circle in red for the reader’s convenience.

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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A Memory Efficient Unified Algorithm for Online Learning of Linear Dynamical Systems

Yuval Ran-Milo, Angelos Assos, Elad Hazan

Tel Aviv University · Princeton University

Dynamical Systems Online System Identification State Space Models

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Motivated by the challenge of stabilizing a general unknown linear dynamical system (LDS) from observations, we study the natural prerequisite of online prediction. Our goal is to achieve sublinear regret with a memory footprint that adapts to the intrinsic complexity of the dynamics rather than the full hidden-state dimension. We focus on the practically central regime of systems with low instability complexity -- eigenvalues outside the real stable interval that do not decay rapidly, together with non-semisimple modes -- potentially embedded in an otherwise stable real spectrum of much higher dimension; we write

k

for this count. This regime is the primary setting in which stabilization is plausible: we show that many systems with high instability complexity cannot be stabilized without exponentially large controls. Thus, prediction is meaningful for stabilization precisely when the instability complexity is small. Within this regime, we introduce a unified online algorithm that handles every LDS (including non-diagonalizable systems with complex or exploding modes) with a learnable parameter count of

\widetilde{O}(k)

. Finally, we prove a lower bound showing that

k

is a valid complexity measure: any filter-based predictor needs at least

k

filters. Experiments corroborate our theory: on a high-dimensional system, our predictor sharply outperforms prior methods at an equal parameter budget.

Abstract

Dynamical Systems

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