High-order accuracy identification framework for stochastic differential equation with Gaussian white noise

Abstract This paper presents a high-order accuracy data-driven identification framework for stochastic differential equation (SDE) with the Gaussian white noise. The framework improves the accuracy in discovering SDE from the data collected at low frequencies. It first approximates the increment of data as a one-step mapping. The truncated terms of the approximation are the higher order terms of the time step of the collected data. Therefore, the mapping can approximate the data collected with large time steps, i.e., low frequencies with high-order accuracy. Then, the framework approximates the mapping as the Fourier series with a high rate of convergence, which is called the double-even extended series. In this way, we can approximate the average change of SDE with high accuracy and do not need knowledge about the change. After determining the coefficients of the Fourier series from data, the new framework finds an accurate approximation of the SDE. We apply the new framework to achieve the identification of an SD oscillator and a glycolytic oscillation. The new framework accurately discovers the SDEs from data collected at low frequencies, e.g., 4 HZ and 2.5 HZ. Identified SDEs can accurately predict stochastic responses. To the best of the authors' knowledge, the frequency is much lower than existing SDE identification methods.