# PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs

arXiv: https://arxiv.org/abs/1909.10145 · HTML: https://mlsanity.com/paper/1909.10145/

## Key facts

- Cited by 118 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2019.

## Built on by (most cited first, 30 of 118)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks](https://mlsanity.com/paper/2406.02917/) (2024)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Physics-Informed Neural Nets for Control of Dynamical Systems](https://mlsanity.com/paper/2104.02556/) (2021)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers](https://mlsanity.com/paper/2207.12800/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Differentiable Multiple Shooting Layers](https://mlsanity.com/paper/2106.03885/) (2021)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs](https://mlsanity.com/paper/2203.05337/) (2022)
- [Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks](https://mlsanity.com/paper/2110.13330/) (2021)
- [Deep neural network enabled corrective source term approach to hybrid analysis and modeling](https://mlsanity.com/paper/2105.11521/) (2021)

## Abstract (arXiv)

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.