Data-Adaptive Learning of Dynamical Systems by Matching Transfer Operators and Invariant Measures

Jul 16, 2026

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Trajectory-based learning for dynamical systems is weakened by noise, sparsity, and chaos, since small pointwise errors can magnify quickly. A transition-statistics approach is presented for system identification by matching how probability mass moves across a data-adaptive state partition. The Perron–Frobenius operator is approximated with a regularized Ulam Markov matrix using continuous partition-of-unity weights, enabling gradient-based learning of a vector field. Training matches invariant measures via stationary eigenvectors and matches full transition matrices, improving long-time reliability on Lorenz-63, Lorenz-96, and a reduced-order NOAA sea-surface temperature forecasting task, particularly under noise and sparse sampling.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

Introduction

Trajectory-based (Lagrangian) identification

Statistical (Eulerian) objectives and invariant measures

Operator-theoretic framing via Perron–Frobenius/Markov matrices

Data-adaptive meshes for operator approximation

arXiv :2607 .00391v1 [math .NA] 1 Jul 2026 Data-Adaptive Learning of Dynamical Systems by Matching Transfer Operators and Invariant Measures Yinong Huang 1 , Jonah Botvinick-Greenhouse2 , and Yunan Yang3 1 Department of Mathematics, North Carolina State University, Raleigh, NC 2 Center for Applied Mathematics, Cornell University, Ithaca, NY 3 Department of Mathematics, Cornell University, Ithaca, NY Abstract Trajectory-based learning of dynamical systems is often fragile in the presence of noise, chaos, or sparse observations, as small pointwise errors can rapidly amplify. We introduce a transition-statistics approach to system identification that learns dynamics by matching the induced motion of probability mass across a data-adaptive mesh. Given trajectory data, we build an unstructured partition of state space and approximate the Perron–Frobenius operator with a regularized Ulam transition matrix. We replace hard cell indicators with continuous, piecewise-smooth partition-of-unity weights, yielding a Markov matrix supporting gradient-based optimization with respect to the parameters of a learned vector field. This enables two related training objectives: matching invariant measures through the stationary eigenvectors of the transition matrices, and matching the full transition matrices to capture transport between regions of state space. Numerical experiments on Lorenz-63, Lorenz-96, and a reduced-order NOAA sea surface temperature forecasting problem show that transition-statistics matching gives more reliable long-time dynamics than pointwise trajectory matching, particularly under measurement noise and sparse sampling. The approach provides a robust operator-theoretic alternative to trajectory-level losses for learning chaotic and partially observed dynamical systems. Keywords— dynamical system identification; transfer operators; Markov matrix matching; invariant measure matching; optimal transport 1 Introduction Data-driven dynamical system identification seeks to infer an evolution law from observed trajectories. This task arises in weather and climate prediction, biological modeling, engineering systems, and reduced-order modeling of complex physical processes. A common approach is to fit a vector field or flow map by comparing simulated trajectories with observed ones. For example, sparse regression methods such as SINDy estimate candidate terms in the governing equations from trajectory data [1], while neural differential equation and shooting-based methods parameterize the vector field and optimize a pointwise trajectory loss [2, 3] . Such trajectory-based, or Lagrangian, objectives can be effective when the data are accurate and sufficiently well sampled. However, they become fragile under noise, sparsity, and chaos which are the regimes considered in this paper. Measurement noise corrupts derivative estimates and can cause a learned model to overfit local fluctuations. Slow sampling makes the one-step map harder to infer and can introduce spurious minima in the optimization landscape. Chaotic dynamics introduce an additional difficulty: even when a model has the correct qualitative behavior, small errors in the vector field or initial condition may lead to rapid pointwise separation of trajectories. Thus, long-time pointwise trajectory agreement is often too stringent a criterion for learning chaotic systems. An alternative is to compare statistical quantities that are stable under the long-time dynamics. Invariant measures provide one such description: rather than recording the precise location of a trajectory at each time, they describe the distribution of states visited asymptotically by the system. For chaotic systems, this perspective is natural because nearby trajectories may separate while still sampling the same invariant measure. This observation has motivated a class of Eulerian system identification frameworks in which the learned dynamics are trained to reproduce global statist

FAQs

Why do pointwise trajectory losses become fragile for chaotic dynamical systems?

Because noise, sparsity, and chaos can cause small errors in the vector field or initial conditions to rapidly separate trajectories. Long-time pointwise agreement is therefore an overly strict learning criterion.

What is the core idea of the proposed transition-statistics approach?

It learns system dynamics by matching induced motion of probability mass across a data-adaptive mesh, approximating the Perron–Frobenius operator with a regularized Ulam Markov matrix.

How are invariant measures and transition dynamics used as training objectives?

Invariant measures are matched through stationary eigenvectors of the transition matrices, while full Markov matrix matching compares transition probabilities to capture transport between regions, not only long-time accumulation.

Jul 16, 2026

Upload by Himbo