Significant progress has been made to obtain approximate solutions to Partial Differential Equations (PDEs) using neural networks as a basis. One of these approaches (the most popular and well-developed one) is the Physics Informed Neural Network (PINN) [1]. PINN has proved to provide promising results in various forward and inverse problems with great accuracy. It carries advantages like mesh-free algorithm, effortlessly combines forward and inverse problem-solving in a single platform. However, a single PINN cannot be employed in its native form for solving problems where the PDE changes its form or when there is a discontinuity in the parameters of PDE across different subdomains. This scenario occurs in various problems across different scientific fields. Heat transfer through heterogeneous systems and conjugate heat transfer are examples of this scenario where thermal conductivity is different for different media, and there exists a discontinuity of thermal conductivity at the interface. Using separate local PINNs for each subdomain and connecting the corresponding solutions by interface conditions is a possible solution. However, this approach demands a high computational burden and memory usage. Here, we present a new method, Transfer Physics Informed Neural Network (TPINN), where one or more layers of PINN across different non-overlapping subdomains are changed, keeping the other layers the same for all the subdomains. The subdomains can be obtained by partitioning the global computational domain or physical subdomains part of the problem definition, which adds to the total computational domain. Solutions from different subdomains are connected via problem-specific interface conditions, which are incorporated into the loss function. Parameter sharing employed here not only reduces parameter space dimension, memory requirements, and computational burden but also increases accuracy. The efficacy of our approach is demonstrated by solving different forward and inverse problems, including classical benchmark problems and problems involving parameter heterogeneity from the heat transfer area.
References
[1] M. Raissi, P. Perdikaris, G.E. Karniadakis (2018) Physics-Informed Neural Networks: A Deep Learning Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations. Journal of Computational Physics 378:686-707