Adversarial and Self-Adaptive Domain Decomposition Physics-Informed Neural Networks (PINNs) for High order Differential Equations with Discontinuities

The technique of solving differential equations using Physics-Informed Neural Networks (PINNs) has received extensive attention and application. However, employing neural networks with continuous activation functions throughout the domain makes it difficult to accurately approximate functions with discontinuities. The original domain decomposition PINNs rely on prior knowledge, as well as manual decomposition of the domain and manual handle of interface conditions between subnetworks. To address this issue, this paper proposes Adversarial and Self-Adaptive Domain Decomposition Physics-Informed Neural Networks (AS-PINNs), which employ adversarial methods to automatically adjust the boundary positions of subdomains and capture different discontinuities. The self-adaptive domain decomposition makes AS-PINNs to be tailored to high order problems by addressing issues arising from the singularity of differentiation at discontinuities.