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Robust reconstruction of the Fokker-Planck equations from time series at different sampling rates

The reconstruction of the Fokker-Planck equations from time series without prior information is still an open problem. Here, we propose a new method to robustly reconstruct different drift and diffusion terms at different sampling rates. Our method is based on the estimation of the transition probability densities for both of the time series and the stochastic differential equation. We approximate the two terms with the Chebyshev series. Without any prior information, our method can recover the orders and coefficients of the underlying polynomial drift and diffusion terms using the synthetic time series generated by four representative models at different sampling rates. The three important factors affecting the reconstructions are the optimal sample size, the orders, and the coefficients of the Chebyshev series, all of which can be totally determined by a given time series.