D3PINNs: A Novel Physics-Informed Neural Network Framework for Staged Solving of Time-Dependent Partial Differential Equations
Xun Yang, Guanqiu Ma, Maohua Ran
TL;DR
D3PINNs address the challenge of dynamically solving time-dependent PDEs by coupling domain-decomposed PINNs with classical time integration. The method first obtains a reliable initial solution using DDPINNs with enhanced interface treatment, then converts the PDE into an ODE by retaining only the time-derivative term and substituting non-time terms with the DDPINNs solution, i.e., $u_t(\mathbf{x},t) \approx -\mathcal{N}[\hat{u},\mathbf{x},t]$, before integrating in time with standard schemes. The key contribution lies in the three-phase framework and the enhanced interface losses that ensure smooth, scalable solutions across subdomains, yielding superior temporal accuracy and robustness compared with existing time-evolving solvers. Numerical experiments on diffusion and Burgers-type problems demonstrate that D3PINNs achieve smaller relative errors and competitive runtimes, highlighting the practical impact for efficient, dynamic PDE solving in structured domains. The work paves the way for applying hybrid PINN-domain-decomposition methods to broader geometries and more complex multiscale PDEs.
Abstract
In this paper, we propose a novel framework, Dynamic Domain Decomposition Physics-Informed Neural Networks (D3PINNs), for solving time-dependent partial differential equations (PDEs). In this framework, solutions of time-dependent PDEs are dynamically captured. First, an approximate solution is obtained by the Physics-Informed Neural Networks (PINNs) containing the domain decomposition, then the time derivative terms in the PDE will be retained and the other terms associated with the solution will be replaced with the approximate solution. As a result, the PDE reduces to an ordinary differential equations (ODEs). Finally, the time-varying solution will be solved by the classical numerical methods for ODEs. D3PINNs retain the computational efffciency and ffexibility inherent to PINNs and enhance the ability for capturing solutions of time-dependent PDEs. Numerical experiments validate the effectiveness of the proposed methods.