A PINN-L1 Method for Inverse Source Estimation in FDEs

We address the use of artificial neural networks (ANNs) to solve partial differential equations (PDEs) with fractional derivatives, specifically based on the Caputo definition. The significance of fractional-order PDEs stems from their application in physical problems such as anomalous heat conduction and polymer behavior modeling. This highlights the need to develop tools to obtain solutions in such cases. In this context, our objective is to develop a methodology based on physics-informed neural networks (PINNs) for solving fractional-order PDEs. The proposed methodology is applied to a case study with a manufactured solution. This allows us to evaluate how this approach can be a promising alternative to numerical methods, especially for complex problems where analytical solutions are challenging to obtain. For the inverse problem, we seek to solve the differential equation with an unknown source function in the differential equation. We use a condition that describes the solution applied in an interval, turning possible estimate the solution and the source function. We consider PINNs build as multilayer perceptrons (MLPs), which are ANNs formed by fully connect layers of basic processing units (perceptrons). Training or calibration involves determining the network’s parameters (weights and bias) so that it yields the estimation of the solution of the goal PDE. Following an backpropagation procedure, the network’s parameters are chosen to minimize an given loss function. It contains the residual and the boundary conditions of the PDE problem. Training stop criteria include a minimum residual value and learning rate. The randomization of the collocation points contribute to avoid the overfitting. The results obtained so far are partial but promising by considering similar applications found in the recent literature and previous works with an unknown term of a source function.