DDC-PINNs: A Predictor-Corrector Approach Based on Neural Network-Driven Domain Decomposition and Classical ODE Solvers for Time-Dependent PDEs

Abstract

When solving time-dependent partial differential equations (PDEs), traditional physics-informed neural networks (PINNs) may encounter several challenges. In particular, standard PINNs do not explicitly account for the temporal evolution order of time-dependent problems during training, which may affect the quality of temporal evolution in time-dependent PDEs. In addition, a single neural network may face difficulties in simultaneously representing different physical behaviors across multiple regions of the computational domain.To address these issues, we propose a domain-decomposition-based causal PINNs (DDC-PINNs) framework. The term causal refers to the fact that the temporal evolution is performed sequentially through classical ordinary differential equation (ODE) integration, thereby respecting the natural temporal ordering of time-dependent PDEs. The proposed framework enhances spatial approximation through domain decomposition and employs a sequential temporal-evolution strategy for time-dependent problems.Within this framework, an approximate solution is first obtained using domain-decomposition PINNs. Subsequently, the time-derivative term in the original PDE is retained, while the remaining solution-dependent terms are replaced by the obtained approximation, thereby transforming the original PDE into an auxiliary ODE system. Classical numerical methods for ODEs are then employed to perform temporal evolution without repeated neural-network optimization. As a result, DDC-PINNs decouples spatial approximation from temporal evolution while preserving the temporal evolution order through sequential ODE integration.Numerical experiments on several benchmark problems demonstrate the effectiveness of the proposed framework and provide proof-of-concept validation of the DDC-PINNs methodology.