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Early warning prediction: Onsager-Machlup vs Schrödinger

Xiaoai Xu, Yixuan Zhou, Xiang Zhou, Jingqiao Duan, Ting Gao

TL;DR

The paper tackles the challenge of predicting critical transitions in high-dimensional systems by coupling manifold learning with data-driven stochastic dynamics and Schrödinger-bridge theory. It introduces two early-warning indicators: an Onsager-Machlup (OM) indicator and a Score Function (SF) indicator derived from the learned score function, with the SF indicator showing superior sensitivity and robustness in epilepsy prediction. The framework is validated on EEG data, including cross-dataset validation, and is supported by a formal high-probability error bound for the SF indicator, highlighting reliable performance under practical conditions. The approach offers a generalizable, data-driven pathway for early warning in diverse complex systems and lays the groundwork for extensions to non-Gaussian noise and broader applications beyond neuroscience.

Abstract

Predicting critical transitions in complex systems, such as epileptic seizures in the brain, represents a major challenge in scientific research. The high-dimensional characteristics and hidden critical signals further complicate early-warning tasks. This study proposes a novel early-warning framework that integrates manifold learning with stochastic dynamical system modeling. Through systematic comparison, six methods including diffusion maps (DM) are selected to construct low-dimensional representations. Based on these, a data-driven stochastic differential equation model is established to robustly estimate the probability evolution scoring function of the system. Building on this, a new Score Function (SF) indicator is defined by incorporating Schrödinger bridge theory to quantify the likelihood of significant state transitions in the system. Experiments demonstrate that this indicator exhibits higher sensitivity and robustness in epilepsy prediction, enables earlier identification of critical points, and clearly captures dynamic features across various stages before and after seizure onset. This work provides a systematic theoretical framework and practical methodology for extracting early-warning signals from high-dimensional data.

Early warning prediction: Onsager-Machlup vs Schrödinger

TL;DR

The paper tackles the challenge of predicting critical transitions in high-dimensional systems by coupling manifold learning with data-driven stochastic dynamics and Schrödinger-bridge theory. It introduces two early-warning indicators: an Onsager-Machlup (OM) indicator and a Score Function (SF) indicator derived from the learned score function, with the SF indicator showing superior sensitivity and robustness in epilepsy prediction. The framework is validated on EEG data, including cross-dataset validation, and is supported by a formal high-probability error bound for the SF indicator, highlighting reliable performance under practical conditions. The approach offers a generalizable, data-driven pathway for early warning in diverse complex systems and lays the groundwork for extensions to non-Gaussian noise and broader applications beyond neuroscience.

Abstract

Predicting critical transitions in complex systems, such as epileptic seizures in the brain, represents a major challenge in scientific research. The high-dimensional characteristics and hidden critical signals further complicate early-warning tasks. This study proposes a novel early-warning framework that integrates manifold learning with stochastic dynamical system modeling. Through systematic comparison, six methods including diffusion maps (DM) are selected to construct low-dimensional representations. Based on these, a data-driven stochastic differential equation model is established to robustly estimate the probability evolution scoring function of the system. Building on this, a new Score Function (SF) indicator is defined by incorporating Schrödinger bridge theory to quantify the likelihood of significant state transitions in the system. Experiments demonstrate that this indicator exhibits higher sensitivity and robustness in epilepsy prediction, enables earlier identification of critical points, and clearly captures dynamic features across various stages before and after seizure onset. This work provides a systematic theoretical framework and practical methodology for extracting early-warning signals from high-dimensional data.
Paper Structure (19 sections, 1 theorem, 28 equations, 1 figure, 4 tables)

This paper contains 19 sections, 1 theorem, 28 equations, 1 figure, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. Background Knowledge
  3. Dimensionality Reduction via Manifold Learning
  4. Onsager-Machlup Action Functional and Schrödinger Bridge Problem
  5. Onsager-Machlup Action Functional
  6. Schrödinger Bridge Problem
  7. Score Matching Method
  8. Early Warning Indicators
  9. Onsager-Machlup Indicator
  10. Score Function Indicator
  11. Comparison of Two Indicators
  12. Experiment
  13. Data Processing and Dimensionality Reduction
  14. Learning SDE in Latent Space
  15. Learning Score Function via Score Matching
  16. ...and 4 more sections

Key Result

Theorem 5.1

Under the above assumptions, for any confidence level $\delta\in(0,1)$, with probability at least $1-\delta$ the estimated Early-warning Potential satisfies the uniform bound where The constants depend only on $S_{\max},L_s,p_{\min},\sigma_{\min},\sigma_{\max},T,l$, and are given explicitly by

Figures (1)

  • Figure 1: Overall Framework Flowchart

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 2.2
  • Theorem 5.1: High-probability error bound for Early-warning Potential
  • proof
  • Remark 5.1