Deep learning-based parameter estimation of stochastic differential equations driven by fractional Brownian motions with measurement noise

This study proposes a general parameter estimation neural network (PENN) to jointly identify the system parameters and the noise parameters of a stochastic differential equation driven by fractional Brownian motion (FBM) from a short sample trajectory. It separately extracts deep features from the trajectory and fuses the information of sampling frequency by a two-stage neural network architecture such that the sample trajectories with variable lengths and sampling times can be properly processed. In addition, by considering additive Gaussian measurement noise in the training stage and utilizing suitable loss functions, the PENN can quantitatively estimate the level of measurement noise and reduce its negative impacts on estimating the governing parameters. Experiments on Fitzhugh–Nagumo model, Duffing oscillator and genetic toggle switch model demonstrate that the PENN can accurately estimate the system parameters, the noise intensity and Hurst exponent of the process noise as well as the signal-to-noise ratio of the measurement noise with high speed.