Multi-output physics-informed neural network for one- and two-dimensional nonlinear time distributed-order models

In this article, a physics-informed neural network based on the time difference method is developed to solve one-dimensional (1D) and two-dimensional (2D) nonlinear time distributed-order models. The FBN- $\theta$, which is constructed by combining the fractional second order backward difference formula (BDF2) with the fractional Newton-Gregory formula, where a second-order composite numerical integral formula is used to approximate the distributed-order derivative, and the time direction at time $t_{n+\frac{1}{2}}$ is approximated by making use of the Crank-Nicolson scheme. Selecting the hyperbolic tangent function as the activation function, we construct a multi-output neural network to obtain the numerical solution, which is constrained by the time discrete formula and boundary conditions. Automatic differentiation technology is developed to calculate the spatial partial derivatives. Numerical results are provided to confirm the effectiveness and feasibility of the proposed method and illustrate that compared with the single output neural network, using the multi-output neural network can effectively improve the accuracy of the predicted solution and save a lot of computing time.