Physics-informed neural networks for Fokker–Planck–Kolmogorov equations corresponding to systems with tempered stable Lévy noise

Tempered stable Lévy noise has a greater advantage in fitting real systems due to its finite variance, compared with α-stable Lévy noise. The responses of a system under tempered stable Lévy noise are governed by a tempered fractional Fokker–Planck-Kolmogorov (FPK) equation, which is difficult to solve due to the nonlocal property of the tempered fractional derivative. In this paper, we propose the tempered fractional physics-informed neural networks (TF-PINNs) for solving the FPK equations corresponding to systems driven by tempered stable Lévy noise. Firstly, we derive the corresponding tempered fractional FPK equation by means of characteristic function, Chapman–Kolmogorov-Smoluchowski equation, and Fourier transformation. Secondly, we extend the PINNs to solve the tempered fractional FPK equation by incorporating discretized tempered fractional derivatives into the loss function of the neural network. In particular, in addition to solving the system response, the proposed TF-PINNs framework can also identify the unknown parameters in the system. Finally, three typical examples are implemented to verify the effectiveness and accuracy of the presented algorithm compared with finite difference solutions and Monte Carlo simulations.