Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

Karniadakis, Ameya D. Jagtap & George Em

doi:10.4208/cicp.oa-2020-0164

Cited by 430 publications

(83 citation statements)

References 30 publications

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“…Jagtap et al [28] develop conservative PINNs (cPINNs) to incorporate complex non-regular geometries by decomposing a spatial domain into independent parts and train separate PINNs for each. This work has been generalised to the extended PINNs (XPINNs) [15] to allow space-time domain decomposition that can be applied to any type of PDE and enables parallelisation in training. Another domain decomposition method for PINNs are the hp-VPINNs [31].…”

Section: Pinns: State Of the Artmentioning

confidence: 99%

Can Physics-Informed Neural Networks beat the Finite Element Method?

Grossmann¹,

Komorowska²,

Latz³

et al. 2023

Preprint

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Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen-Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.

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Section: Pinns: State Of the Artmentioning

confidence: 99%

Can Physics-Informed Neural Networks beat the Finite Element Method?

Grossmann¹,

Komorowska²,

Latz³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…This directly relates to one of the main branch of physically-guided neural networks that aims at designing specific NN architectures to embed the physics in the modeling system. An other option would be to keep similar CNN and attention-based architectures proposed in the paper while adding additional constraints on the physics in the loss function: it is an active field of research in what is called 'physically-informed neural networks' [46][47][48]. At last, the use of satellite data like aerosol optical depth with ground observations as input data for a NN allows to create an adequate model to predict Super-Resolution PM2.5 concentrations as it has been reported over Beijing [49,50].…”

Section: Discussionmentioning

confidence: 99%

Deep learning techniques applied to super-resolution chemistry transport modeling for operational uses

Bessagnet

Beauchamp

Menut

et al. 2021

Environ. Res. Commun.

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Air quality modeling tools are largely used to assess air pollution mitigation and monitoring strategies. While neural networks (NN) were mostly developed based on observations to derive statistical models at stations, the use of Eulerian chemistry transport models (CTMs) was mainly devoted to air quality predictions over large areas and the evaluation of emission reduction strategies. In this study, we investigate deep learning architectures to create a metamodel of the process oriented CTM CHIMERE and significantly reduce the computing times required for super-resolution simulations. The key point is the selection of input variables and the way to implement them in the NN. We perform a quantitative evaluation of the proposed approaches on a real case-study. The best NN architecture displays very good performances in terms of prediction of pollutant concentrations observed at stations with respect to the raw super-resolution CHIMERE simulation, with a correlation coefficient above 0.95. The best NN is also able to display better performances when compared to observations than the raw high resolution simulation. Currently the model is designed to be used for air quality forecasting and requires improvement for the definition of air quality management strategies.

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“…The general framework was presented by Parish et al in [39]. The idea is currently extensively studied as can be seen in [40][41][42] and used in complex applications [43,44]. An advanced approach in the field of Computational Fluid Dynamics (CFD) was proposed by Ling et al [45] and later on by Wu et al in [46].…”

Section: Literature Reviewmentioning

confidence: 99%

Integration of Classical Mathematical Modeling with an Artificial Neural Network for the Problems with Limited Dataset

2021

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One of the most common problems in science is to investigate a function describing a system. When the estimate is made based on a classical mathematical model (white-box), the function is obtained throughout solving a differential equation. Alternatively, the prediction can be made by an artificial neural network (black-box) based on trends found in past data. Both approaches have their advantages and disadvantages. Mathematical models were seen as more trustworthy as their prediction is based on the laws of physics expressed in the form of mathematical equations. However, the majority of existing mathematical models include different empirical parameters, and both approaches inherit inevitable experimental errors. Simultaneously, the approximation of neural networks can reproduce the solution exceptionally well if fed sufficient data. The difference is that an artificial neural network requires big data to build its accurate approximation, whereas a typical mathematical model needs several data points to estimate an empirical constant. Therefore, the common problem that developers meet is the inaccuracy of mathematical models and artificial neural networks. Another common challenge is the mathematical models’ computational complexity or lack of data for a sufficient precision of the artificial neural networks. Here we analyze a grey-box solution in which an artificial neural network predicts just a part of the mathematical model, and its weights are adjusted based on the mathematical model’s output using the evolutionary approach to avoid overfitting. The performance of the grey-box model is statistically compared to a Dense Neural Network on benchmarking functions. With the use of Shaffer procedure, it was shown that the grey-box approach performs exceptionally well when the overall complexity of a problem is properly distributed with the mathematical model and the Artificial Neural Network. The obtained calculation results indicate that such an approach could increase precision and limit the dataset required for learning. To show the applicability of the presented approach, it was employed in modeling of the electrochemical reaction in the Solid Oxide Fuel Cell’s anode. Implementation of a grey-box model improved the prediction in comparison to the typically used methodology.

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Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

Cited by 430 publications

References 30 publications

Can Physics-Informed Neural Networks beat the Finite Element Method?

Can Physics-Informed Neural Networks beat the Finite Element Method?

Deep learning techniques applied to super-resolution chemistry transport modeling for operational uses

Integration of Classical Mathematical Modeling with an Artificial Neural Network for the Problems with Limited Dataset

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