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

Meng, Xuhui; Li, Zhen; Zhang, Dongkun; Karniadakis, George Em

doi:10.48550/arxiv.1909.10145

Cited by 4 publications

(6 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Physics Informed Neural networks constitute a class of neural networks dedicated to the resolution of PDEs [11]. Consider problem (10), and let us denote by h a fully connected neural network of L hidden layers of I neurons with differentiable activation functions o, and by θ the parameters of our network. We want to train the network so that…”

Section: About Pinnmentioning

confidence: 99%

“…In the short term, improvements can be made, for example, the use of a standard coarse numerical solver instead of a PINN is a promising avenue and brings us closer to the work of [10] (without however decoupling the problems between them). We also wish to apply the principle of PINNs to the residuals of the physical problem as is done in the classical literature [1] and strengthen the interface conditions using the work already done in [6] and [4].…”

Section: Future Work and Possible Improvementsmentioning

confidence: 99%

See 1 more Smart Citation

A coarse space acceleration of deep-DDM

Valentin¹,

Gratton²,

Boudier³

2021

Preprint

View full text Add to dashboard Cite

The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the partial differential equation, have shown their great potential. Yet, to address large scale problems encountered in real applications and compete with existing numerical methods for PDEs, it is important to design parallel algorithms with good scalability properties. In the vein of traditional domain decomposition methods (DDM), we consider the recently proposed deep-ddm approach. We present an extension of this method that relies on the use of a coarse space correction, similarly to what is done in traditional DDM solvers. Our investigations shows that the coarse correction is able to alleviate the deterioration of the convergence of the solver when the number of subdomains is increased thanks to an instantaneous information exchange between subdomains at each iteration. Experimental results demonstrate that our approach induces a remarkable acceleration of the original deep-ddm method, at a reduced additional computational cost.

show abstract

Section: About Pinnmentioning

confidence: 99%

Section: Future Work and Possible Improvementsmentioning

confidence: 99%

A coarse space acceleration of deep-DDM

Valentin¹,

Gratton²,

Boudier³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Another approach to using neural networks in a parareal framework is discussed in [18]. The authors propose parareal physics-informed neural network (PPINN).…”

Section: Other Related Workmentioning

confidence: 99%

“…See [22]. The key idea of [18] is using the the parareal scheme to train the sequence of PINNs in parallel. However, to the best of our current knowledge, the accuracy and stability of the PINN method for wave problems are not well understood.…”

Section: Other Related Workmentioning

confidence: 99%

Numerical wave propagation aided by deep learning

Nguyen¹,

Tsai²

2021

Preprint

View full text Add to dashboard Cite

We propose a deep learning approach for wave propagation in media with multiscale wave speed, using a second-order linear wave equation model. We use neural networks to enhance the accuracy of a given inaccurate coarse solver, which under-resolves a class of multiscale wave media and wave fields of interest. Our approach involves generating training data by the given computationally efficient coarse solver and another sufficiently accurate solver, applied to a class of wave media (described by their wave speed profiles) and initial wave fields. We find that the trained neural networks can approximate the nonlinear dependence of the propagation on the wave speed as long as the causality is appropriately sampled in training data. We combine the neuralnetwork-enhanced coarse solver with the parareal algorithm and demonstrate that the coupled approach improves the stability of parareal algorithms for wave propagation and improves the accuracy of the enhanced coarse solvers.

show abstract

“…However, vanilla GPR has difficulties in handling the nonlinearities when applied to solve PDEs, leading to restricted applications. On the other hand, PINNs have shown effectiveness in both forward and inverse problems for a wide range of PDEs [9,10,11,12,13]. However, PINNs are not equipped with built-in uncertainty quantification, which may restrict their applications, especially for scenarios where the data are noisy.…”

Section: Introductionmentioning

confidence: 99%

B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data

Yang,

Meng,

Karniadakis

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINN posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Loève (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.

show abstract

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

Cited by 4 publications

References 20 publications

A coarse space acceleration of deep-DDM

A coarse space acceleration of deep-DDM

Numerical wave propagation aided by deep learning

B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data

Contact Info

Product

Resources

About