Maohua Ran

Haodong Pu, Maohua Ran

This work introduces a novel structure-preserving methods for conservative systems based on a predictor-corrector strategy. The framework applies a discrete gradient correction to predictions generated by explicit one-step or multi-step schemes, which preserves nonlinear invariants while maintaining the accuracy order of the original predictor. This approach extends naturally to problems requiring simultaneous conservation of multiple invariants. Under mild conditions, conservation properties, solvability, numerical accuracy, and stability are established. Long-term numerical simulations on Lotka-Volterra systems, sine-Gordon equations, rigid body dynamics, and Kepler problems demonstrate improved robustness and conservation properties compared to existing projection and relaxation methods.

Xun Yang, Guanqiu Ma, Maohua Ran

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.