udPINNs: An Enhanced PDE Solving Algorithm Incorporating Domain of Dependence Knowledge

Nanxi Chen, Jiyan Qiu, Pengcheng Shi, Wu Yuan, Jian Zhang

Abstract: Physics-Informed Neural Networks (PINNs) have demonstrated their effectiveness in solving partial differential equations (PDEs) by integrating PDE knowledge into the neural network training process. However, prior methods were restricted to incorporating only PDE knowledge, and could not utilize broader knowledge from computational mathematics, such as the domain of dependence of PDEs. To tackle this limitation, we introduce a distributed PINNs algorithm called udPINNs (unidirectional Physics-Informed Neural Networks), which is founded on a domain decomposition approach and capable of incorporating domain of dependence knowledge into the training process. This enhancement accelerates training speed and elevates solution accuracy. We validate udPINNs on common equations, including heat transfer and incompressible flow, and demonstrate that it surpasses existing XPINNs methods in terms of error reduction and computational efficiency.