Early warning prediction: Onsager-Machlup vs Schrödinger

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.

💡 Research Summary

The paper presents a comprehensive framework for early‑warning prediction of critical transitions in complex, high‑dimensional systems, with a focus on epileptic seizures recorded by electroencephalography (EEG). The authors first address the curse of dimensionality by applying manifold learning to compress the raw multichannel EEG data into a low‑dimensional representation. Six representative dimensionality‑reduction techniques are evaluated: Orthogonal Locality Preserving Projection (OLPP), Diffusion Maps (DM), Isomap, Principal Component Analysis (PCA), Kernel PCA, and Neighborhood Preserving Embedding (NPE). Quantitative comparisons of reconstruction error, preservation of global and local geometry, and computational cost reveal that Diffusion Maps best retain the intrinsic nonlinear structure of the data, making it the most suitable choice for subsequent modeling.

On the reduced manifold, a data‑driven stochastic differential equation (SDE) model is constructed: dX(t)=b(X(t))dt+σ(X(t))dW(t), where b(·) denotes a nonlinear drift and σ(·) a state‑dependent diffusion matrix. The drift and diffusion functions are estimated using a hybrid of maximum‑likelihood and score‑matching techniques, ensuring that the low‑dimensional stochastic dynamics faithfully reproduce statistical properties of the original high‑dimensional signals.

Two early‑warning indicators are then derived from this stochastic model. The first indicator is based on the classical Onsager‑Machlup (OM) action functional. The OM functional assigns a probability density to an entire trajectory z(t) via exp(−S_OM), where S_OM = ∫₀ᵀ