Non-local physics informed neural networks for forward and inverse problems containing non-local operators

Non-local physics informed neural networks for forward and inverse problems containing non-local operators

연구 분야: Artificial Intelligence

학회: Neural Computing and Applications

A novel idea, physics informed neural networks introduced a few years back to solve forward and inverse problems for differential equations using the physics information that lies inside them. The central pillar of the physics informed neural networks is automatic differentiation, which is based on the chain rule of differentiation. Automatic differentiation is not applicable to non-local operators because the standard chain rule of differentiation is not valid for non-local operators. Therefore, this work presents non-local physics informed neural networks, which use standard approximation methods for non-local operators and automatic differentiation for local operators to solve differential equations containing non-local operators (forward problems) as well as learn differential equations involving non-local operators (inverse problems). In this work, we consider the Caputo fractional derivative, Volterra integral, and Itô integral as non-local operators. Moreover, we demonstrate the efficiency of the non-local physics informed neural networks with different test examples like the time-fractional diffusion equation in one and two dimensions, the time-fractional Burgers’ equation (both equations involving Caputo fractional derivative as non-local operators), the fractional integro-differential equation (Caputo fractional derivative and Volterra integral as non-local operators), and the stochastic fractional integro-differential equation (Caputo fractional derivative, Itô integral, and Volterra integral as non-local operators). Furthermore, for the non-smooth solution, we use the approximation method on non-uniform mesh for non-local operators and compare the results with the approximation method on uniform mesh. We also discuss the error analysis and convergence of the proposed non-local physics informed neural networks. Finally, we take real-world data, which is described by the differential equation containing non-local operators, and show the effectiveness of non-local physics-informed neural networks in addressing practical applications.