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

## TL;DR

DDC-PINNs combines domain decomposition, causality constraints, and classical ODE solvers to improve the accuracy and efficiency of neural network solutions for time-dependent PDEs.

## Contribution

It introduces a novel predictor-corrector framework that enhances PINNs with domain decomposition and classical ODE solvers for better handling of time-dependent PDEs.

## Key findings

- Improved accuracy in solving time-dependent PDEs.
- Effective incorporation of causality constraints.
- Numerical experiments demonstrate superior performance.

## Abstract

When solving time-dependent partial differential equations(PDEs), traditional physics-informed neural networks (PINNs) have inherent limitations: due to the lack of temporal causality, the network is forced to learn the later-time control equations while fully capturing the initial conditions, resulting in the continuous accumulation of errors during the integration process. Meanwhile, the limited expressivity of a single network hinders its ability to capture diverse physical behaviors across multiple subdomains. To address these issues, we propose a domain-decomposition-based causal PINNs (DDC-PINNs) framework. This framework enhances spatial representation through domain decomposition and employs a causal strategy to constrain the temporal learning sequence, thereby improving the accuracy and generalization ability of solutions to time-varying problems. Within this framework, an approximate solution is first obtained through PINNs with domain decomposition. Subsequently, the time derivative term in the PDE is retained, while other solution-dependent terms are replaced with this approximate solution, thereby simplifying the original PDEs into ordinary differential equations (ODEs). Finally, classical numerical methods for solving ODEs are employed to obtain the time-dependent solution. DDC-PINNs not only preserve the inherent computational efficiency and flexibility of PINNs but also effectively incorporate causality when solving time-dependent PDEs. Numerical experiments verify the effectiveness of the proposed method.