A deep learning solution of time-fractional Fokker–Planck equation for special stochastic processes in foreign exchange market

The earnings behavior of the foreign exchange market at different time scales can be described by time-fractional Fokker–Planck equation (TFFPE). In order to solve this equation, we propose a deep learning method, called Sumudu transform physics-informed neural networks (ST-PINNs). ST-PINNs adopts Sumudu transform and Newton interpolation to approximate the fractional derivatives of the neural network output which changes the limitation that PINNs cannot be directly applied to solve fractional partial differential equations. In addition, we design appropriate neural networks and new loss functions to enable ST-PINNs to solve TFFPE better. To verify the effectiveness of this method, we present numerical cases of solving 1D, 2D and 3D TFFPEs when 0 < α < 2 . The results show that both absolute error and relative L 2 error can be within 1 × 1 0 − 2 , and even noise is considered, our method can still achieve very good results.