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Data-driven Discovery of Invariant Measures

Jason J. Bramburger, Giovanni Fantuzzi

TL;DR

The paper tackles recovering invariant measures that govern long-time dynamical behavior from data, without explicit dynamics, by formulating a convex optimization over moments. It leverages a data-driven EDMD approximation of the Koopman/Lie derivative to build a data-driven SDP that targets physical or ergodic measures, including singular ones, via $A_k y=0$ and $M(\sigma_j y)\succeq0$ constraints. Convergence is established for polynomial maps, while showcased on the logistic map, a stochastic double-well, and the Rössler attractor, demonstrating ability to recover attractor statistics and extract unstable periodic orbits from a Poincaré map. Compared to Ulam's method and model-based EDMD, this approach is more scalable and directly targets extremal invariant measures, enabling robust data-driven discovery of UPOs and attractor statistics.

Abstract

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.

Data-driven Discovery of Invariant Measures

TL;DR

The paper tackles recovering invariant measures that govern long-time dynamical behavior from data, without explicit dynamics, by formulating a convex optimization over moments. It leverages a data-driven EDMD approximation of the Koopman/Lie derivative to build a data-driven SDP that targets physical or ergodic measures, including singular ones, via and constraints. Convergence is established for polynomial maps, while showcased on the logistic map, a stochastic double-well, and the Rössler attractor, demonstrating ability to recover attractor statistics and extract unstable periodic orbits from a Poincaré map. Compared to Ulam's method and model-based EDMD, this approach is more scalable and directly targets extremal invariant measures, enabling robust data-driven discovery of UPOs and attractor statistics.

Abstract

Invariant measures encode the long-time behaviour of a dynamical system. In this work, we propose an optimization-based method to discover invariant measures directly from data gathered from a system. Our method does not require an explicit model for the dynamics and allows one to target specific invariant measures, such as physical and ergodic measures. Moreover, it applies to both deterministic and stochastic dynamics in either continuous or discrete time. We provide convergence results and illustrate the performance of our method on data from the logistic map and a stochastic double-well system, for which invariant measures can be found by other means. We then use our method to approximate the physical measure of the chaotic attractor of the Rössler system, and we extract unstable periodic orbits embedded in this attractor by identifying discrete-time periodic points of a suitably defined Poincaré map. This final example is truly data-driven and shows that our method can significantly outperform previous approaches based on model identification.
Paper Structure (26 sections, 6 theorems, 51 equations, 10 figures, 3 tables)

This paper contains 26 sections, 6 theorems, 51 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Invariant Measures
  3. Background and Definitions
  4. Convex Computation of Invariant Measures
  5. Moment-based Characterization of Invariance
  6. Optimization over Invariant Measures
  7. Physical Measures.
  8. Ergodic Measures.
  9. Approximation via Semidefinite Programming
  10. Recovery of an Approximate Invariant Measure
  11. Discovering Extremal Invariant Measures from Data
  12. Convergence for Polynomial Maps
  13. Preliminaries
  14. Almost-sure Strict Feasibility
  15. Convergence with Infinite Data
  16. ...and 11 more sections

Key Result

Theorem 2.1

Suppose ass:poly-fass:semiagebraic-X hold. Let $y^k \in \mathbb{R}^{\binom{n+kd}{kd}}$ be the optimal solution of MeasureSDP for fixed $k \in \mathbb{N}$, and let $\mu$ be optimal for MeasureLP. There exists an increasing sequence $\{k_j\}_{j \geq 1} \subset \mathbb{N}$ such that In particular, if MeasureLP has a unique optimizer $\mu$, then $y_\alpha^k \to \int x^\alpha \,\mathrm{d}\mu$ as $k\to

Figures (10)

  • Figure 1: Feasible sets for the SDPs \ref{['e:example-failure-exact']} (left) and \ref{['e:example-failure-data']} (right). In both panels, the orange shaded region is the set of points satisfying the (matrix) inequality constraints. Straight black lines indicate the linear spaces $y_1 - y_2=0$ (left panel) and $\alpha_m y_1 - \beta_m y_2 = 0$ for $\alpha_m/\beta_m > 1$ (right panel). Optimal points in both panels are marked by a blue dot.
  • Figure 2: Discovered densities (blue, solid) for the logistic map \ref{['Logistic']} using $m = 10^4$ data points compared to the exact density \ref{['LogisticDensity']} (red, dashed).
  • Figure 3: Left: Histogram approximation (blue) of the density of the physical measure \ref{['LogisticDensity']} of the logistic map against the exact density (red, dashed). We use $m = 10^3$ datapoints along a single trajectory starting with $x_0 = 0.25$ and organize data into 101 bins. Right: The approximated density using the SDP methods of this manuscript (blue) against the true density (red, dashed) using the same datapoints as the left panel and taking $k = 20$.
  • Figure 4: Polynomial densities of Lebesgue-absolutely-continuous measures that approximates the atomic invariant measure supported on the fixed point $x = -0.5$ of the logistic map \ref{['Logistic']}. Results are for $m = 10^4$ data snapshots and polynomial degrees $k=5$, $10$ and $20$.
  • Figure 5: Left-most panel: Invariant density $\rho$ for the stochastic system in \ref{['e:DoubleWell']} for with $\sigma=0.75$. The formula for $\rho$ is given by \ref{['e:double-well-exact-sol']}. Other panels: Approximations to $\rho$ obtained from the dataset $\mathcal{D}_1$ with (in order) with our SDP approach, with a data-based histogram, and with a two-component Gaussian mixture fit to the data. In each panel, dotted white lines mark the reflection symmetry axes ($x_2=x_1$ and $x_2=-x_1$) for the exact invariant density.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Theorem 2.1: Korda2021invariant
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Theorem 4.2: bramburger2023auxiliary
  • Proposition 4.1
  • proof
  • Theorem 4.3
  • Theorem 4.3
  • proof
  • ...and 1 more