Tracing back transient information from near-stationary random data

Tracing back the past and predicting the future are of equal importance, while compared to the prediction, the backtracking is far from receiving the attention it deserves. With the explosive advances of the diffusion models, backtracking has undergone a complete renaissance, especially for stochastic systems. This work addresses this issue, tracing back transient information from near-stationary random data. Different from the diffusion models, we aim for statistical information but not sample information, and we only need near-stationary sample segments for identification, but not a large number of full-time samples for learning. The core idea of this data-driven method is: embedding Fokker-Planck equation (as a priori physical knowledge), which portrays the evolution of probability density of state, and then identifying and solving it to trace back the transient probability density. The efficacy of this method is demonstrated by three typical examples, namely, a one-dimensional linear system, a two-dimensional linear system, and the van der Pol system.