Deep Learning Nonhomogeneous Elliptic Interface Problems by Soft Constraint Physics-Informed Neural Networks

It is a great challenge to solve nonhomogeneous elliptic interface problems, because the interface divides the computational domain into two disjoint parts, and the solution may change dramatically across the interface. A soft constraint physics-informed neural network with dual neural networks is proposed, which is composed of two separate neural networks for each subdomain, which are coupled by the connecting conditions on the interface. It is beneficial to capture the singularity of the solution across the interface. We formulate the PDEs, boundary conditions, and jump conditions on the interface into the loss function by means of the physics-informed neural network (PINN), and the different terms in the loss function are balanced by optimized penalty weights. To enhance computing efficiency for increasingly difficult issues, adaptive activation functions and the adaptive sampled method are used, which may be improved to produce the optimal network performance, as the topology of the loss function involved in the optimization process changes dynamically. Lastly, we present many numerical experiments, in both 2D and 3D, to demonstrate the proposed method's flexibility, efficacy, and accuracy in tackling nonhomogeneous interface issues.