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Learning effective dynamics from data-driven stochastic systems

Lingyu Feng, Ting Gao, Min Dai, Jinqiao Duan

TL;DR

The paper tackles the challenge of learning effective dynamics for unknown slow-fast stochastic systems from limited, short-term data by introducing Auto-SDE, a time-evolving autoencoder that simultaneously estimates the governing SDE via Kramers–Moyal relations and propagates reduced dynamics on an inferred invariant slow manifold. Leveraging Fenichel theory, the method aims to recover a reduced, low-dimensional representation that captures long-term behavior while remaining faithful to the original high-dimensional stochastic dynamics. Numerical experiments on nonlinear slow-fast systems show that Auto-SDE can accurately approximate the invariant manifold and reproduce the trajectories and distributions of the slow variables, with performance robust to noise levels. This framework enables reliable long-term predictions from sparse data and offers a practical tool for data-driven multiscale modeling in science and engineering.

Abstract

Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective dynamics for slow-fast stochastic dynamical systems. Given observation data on a short-term period satisfying some unknown slow-fast stochastic systems, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable and effective through numerical experiments under various evaluation metrics.

Learning effective dynamics from data-driven stochastic systems

TL;DR

The paper tackles the challenge of learning effective dynamics for unknown slow-fast stochastic systems from limited, short-term data by introducing Auto-SDE, a time-evolving autoencoder that simultaneously estimates the governing SDE via Kramers–Moyal relations and propagates reduced dynamics on an inferred invariant slow manifold. Leveraging Fenichel theory, the method aims to recover a reduced, low-dimensional representation that captures long-term behavior while remaining faithful to the original high-dimensional stochastic dynamics. Numerical experiments on nonlinear slow-fast systems show that Auto-SDE can accurately approximate the invariant manifold and reproduce the trajectories and distributions of the slow variables, with performance robust to noise levels. This framework enables reliable long-term predictions from sparse data and offers a practical tool for data-driven multiscale modeling in science and engineering.

Abstract

Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective dynamics for slow-fast stochastic dynamical systems. Given observation data on a short-term period satisfying some unknown slow-fast stochastic systems, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable and effective through numerical experiments under various evaluation metrics.
Paper Structure (11 sections, 16 equations, 15 figures, 3 tables, 1 algorithm)

This paper contains 11 sections, 16 equations, 15 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Slow manifold
  4. Reduced system on invariant manifold
  5. Methods
  6. Recovery of the slow-fast system
  7. Time evolutionary auto-encoder neural network
  8. Loss function and training algorithm
  9. Numerical experiments
  10. Conclusion and discussion
  11. Appendix

Figures (15)

  • Figure 1: A schematic of our framework and the workflow during the training phase. (a) The auxiliary estimation network to identify the stochastic differential equations. (b) Auto-SDE consists of a 3-layer encoder, a LSTM layer and a 3-layer decoder. The Auto-SDE network adopts overlapping technique with reconstruction and stochastic differential equations to predict time evolution on the lower dimensional representation effectively.
  • Figure 2: The input dataset (blue dot) and the approximated manifold (red curve).
  • Figure 3: The snapshots of 1200 initial points evolving over time. The mark $NT$ represents the Nth time step with step size $\Delta t =0.001$.
  • Figure 4: The trajectories generated by the approximated reduced system of our network and the slow variable of the original system. The starting point is on the approximated manifold.
  • Figure 5: The distribution of 1000 trajectories generated by the approximated reduced system of our network and the slow variable of the original system. From left to right, the value of $NT$ is equal to 10, 100, or 1000.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Definition 1
  • Example 1
  • Example 2
  • Example 3