ITPINN : Integral-trainable physics informed neural network for solving high-dimensional evolution non-local partial differential equations

Abstract

In this paper, we develop a new physics informed neural network (PINN) method, named integral-trainable PINN (ITPINN), to solve high-dimensional non-local partial differential equations (PDEs), involving PDEs with fractional derivatives (Caputo derivative and Riemann-Liouville derivative) or multiple integral. In the ITPINN framework, we perform integration by parts on the original integral to obtain a new constraint condition, which forms a coupled system with the original equation. We consider the integral terms as unknown functions in the coupled system and construct a neural network with three output terms, one for predicting the exact solution, one for predicting the original integral term, and one for approximating the new integral obtained by integration by parts. The network is used as a surrogate model for fractional derivatives or multiple integral, which allows approximation of the fractional derivatives or multiple integral to be achieved by training the network. The proposed method omits the process of discretizing the integral term using traditional numerical methods, such as finite difference method or interpolation approximation. Moreover, the physical information obtained from integration by parts is used to construct a new supervised learning task to further constrain the surrogate model for the integral terms. Several experiments are used to illustrate the performance of the ITPINN. The numerical results confirm that our proposed method can effectively solve high-dimensional evolution non-local PDEs, such as 50D problems. Compared to fractional PINN (fPINN) and auxiliary PINN (A-PINN), the ITPINN can achieve higher prediction accuracy and save more training time. In particular, we also test the robustness of the ITPINN under interference with noise intensities ranging from 0.01% to 50% and further discuss its scalability in 100D and 1000D problems.