Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

TL;DR

The paper addresses accelerating time-dependent Allen-Cahn simulations by embedding operator-learning CNNs into the Parareal parallel-in-time framework. By training ACNN and mACNN to approximate the fully discrete time-stepping operator, the authors obtain a consistent, fast coarse propagator for both classic and mass-conservative AC equations. Numerical experiments in 2D and 3D demonstrate rapid Parareal convergence and substantial speedups (up to 4–6× in 2D and around 3× in 3D) on multi-GPU systems, while maintaining high accuracy near the fine-solver results. The approach generalizes across varied initial conditions without retraining, preserves mass in the conservative formulation, and highlights the potential of neural operators within time-parallel frameworks for efficient, accurate simulations of time-dependent PDEs.

Abstract

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned solution as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. To ensure such consistency, we propose to use the convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the traditional numerical scheme used as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen-Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNNs model is trained on trajectories-based on random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without re-training. This work demonstrates the potential of integrating neural network methods into the parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

Figures (22)

• Figure 1: The structure of ACNN (top) and mACNN (bottom) which learns the dynamics of classic AC equation and mass-conservative AC equation respectively.
• Figure 2: Illustration of the Parareal algorithm. The coarse propagator here is labeled $\mathcal{G}$, whereas the fine propagator is labeled $\mathcal{F}$. The red font means that solutions are already the fine solver simulations.
• Figure 3: 2D CNN models: Mean predicted errors and one-standard-deviation bands across ten independently trained models. The left panel shows results for ACNN, while the right panel presents results for mACNN.
• Figure 4: 3D CNN models: Mean predicted errors and one-standard-deviation bands across ten independently trained models. The left panel corresponds to ACNN, and the right panel to mACNN.
• Figure 5: Classic AC in 2D: (Left) The $\|U^{k}-U^{k-1}\|_{\infty}$ with iteration index k and (right) the corresponding relative errors after the $k$-th iteration. Note $k=0$ means the coarse propagator errors without Parareal algorithm.