PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs (1909.10145v1)

Abstract: Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree. While effective for relatively short-term time integration, when long time integration of the time-dependent PDEs is sought, the time-space domain may become arbitrarily large and hence training of the neural network may become prohibitively expensive. To this end, we develop a parareal physics-informed neural network (PPINN), hence decomposing a long-time problem into many independent short-time problems supervised by an inexpensive/fast coarse-grained (CG) solver. In particular, the serial CG solver is designed to provide approximate predictions of the solution at discrete times, while initiate many fine PINNs simultaneously to correct the solution iteratively. There is a two-fold benefit from training PINNs with small-data sets rather than working on a large-data set directly, i.e., training of individual PINNs with small-data is much faster, while training the fine PINNs can be readily parallelized. Consequently, compared to the original PINN approach, the proposed PPINN approach may achieve a significant speedup for long-time integration of PDEs, assuming that the CG solver is fast and can provide reasonable predictions of the solution, hence aiding the PPINN solution to converge in just a few iterations. To investigate the PPINN performance on solving time-dependent PDEs, we first apply the PPINN to solve the Burgers equation, and subsequently we apply the PPINN to solve a two-dimensional nonlinear diffusion-reaction equation. Our results demonstrate that PPINNs converge in a couple of iterations with significant speed-ups proportional to the number of time-subdomains employed.

• The paper introduces PPINN, a novel framework that couples a coarse-grained solver with parallel fine PINNs to efficiently address long-time PDE challenges.
• It demonstrates robust convergence and superlinear speedup in experiments with PDEs like the Burgers equation and nonlinear diffusion-reaction systems.
• The approach employs a prediction-correction methodology that significantly reduces computational costs, enabling scalable simulations for complex physical phenomena.

An Overview of the Parareal Physics-Informed Neural Network for Solving Time-dependent PDEs

The paper "PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs" presents a novel computational framework aimed at addressing the computational challenges associated with the long-time integration of time-dependent partial differential equations (PDEs). The authors propose a method that significantly improves the efficiency of traditional Physics-Informed Neural Networks (PINNs) by incorporating a parareal algorithmic approach, hereafter referred to as Parareal PINN (PPINN).

Framework and Methodology

PPINNs are designed to alleviate the prohibitive computational costs of PINNs when extended to long-time problems by decomposing these into a series of short-time problems. This is achieved using a coarse-grained (CG) solver, which works as a prediction mechanism over larger temporal intervals, and multiple fine PINNs computed in parallel, which serve to iteratively correct the solution. The PPINN approach hence benefits from increased speed through small data set handling by each individual PINN instance, complemented by the ability for parallel computation, which further optimizes computational efficiency.

The methodology follows a systematic prediction-correction framework:

1. Model Reduction: A simplified PDE is solved by the CG solver.
2. Initialization: The CG solver provides an initial prediction over the entire time domain, which serves as input for the fine PINNs.
3. Correction and Refinement: Fine PINNs refine the solution in parallel followed by further iterations that blend corrections into the coarse grained initial solutions, assuring convergence.

Numerical Experiments and Results

The paper reports an experimental evaluation of PPINN using a range of PDE instances, including the one-dimensional Burgers equation and a two-dimensional nonlinear diffusion-reaction system. Key findings include:

• Convergence Efficiency: PPINNs demonstrate an ability to converge within only a few iterations, effectively capturing the true solution dynamics.
• Computational Speed-up: The approach achieved significant speed-ups, with evidence suggesting a superlinear increase in speed as parallelizability is enhanced via decomposition into smaller sub-tasks handled by each PINN.
• Robustness Across Instances: Regardless of PDE complexity, the parareal algorithm maintains performance robustness by using an appropriately reduced order model as the coarse scale solver.

PPINNs have shown favorable scalability with respect to the size of the time domain, demonstrating their potential application to large-scale physical and engineering phenomena simulated via long-time PDEs. The efficiency gains are particularly pronounced when a reliable coarse solver provides initial conditions close to the solution, thereby accelerating the convergence of the finer solver iterations.

Implications and Future Directions

The PPINN approach suggests a substantial opportunity for the development of efficient, scalable solutions to complex PDEs, especially in contexts necessitating long time integrations, such as weather forecasting, fluid dynamics, and materials science. The method's reliance on CG solvers also opens the door for multi-fidelity modeling applications where computational resources differ widely across scales and resolutions.

Looking forward, further improvements could investigate optimal selection criteria for the coarse solver to balance prediction accuracy with computational load. Additionally, extending PPINNs to include spatial domain decomposition could address large spatial data sets, employing principles akin to multigrid or multi-resolution methods.

In conclusion, the PPINN framework marks a promising direction within computational science, leveraging the benefits of parallelization and model order reduction to efficiently handle the inherent challenges of time-dependent PDEs.