Thermodynamically consistent physics-informed neural networks for hyperbolic systems

Patel, Ravi G.; Manickam, Indu; Trask, Nathaniel; Wood, Mitchell; Lee, Myoungkyu; Tomaš, Ignacio; Cyr, Eric C

doi:10.1016/j.jcp.2021.110754

Cited by 80 publications

(22 citation statements)

References 57 publications

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“…The other applications of PINN are discussed briefly. Patel et al [37] investigated thermodynamically consistent PINN for hyperbolic shock hydrodynamics systems. The forward and inverse problems related to the nonlinear diffusivity and Biot's equation were investigated by Kadeethum et al [38].…”

Section: Surrogate Modelmentioning

confidence: 99%

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Das¹,

Tesfamariam²

2022

Preprint

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This paper presents a comprehensive review of the design of experiments used in the surrogate models. In particular, this study demonstrates the necessity of the design of experiment schemes for the Physics-Informed Neural Network (PINN), which belongs to the supervised learning class. Many complex partial differential equations (PDEs) do not have any analytical solution; only numerical methods are used to solve the equations, which is computationally expensive. In recent decades, PINN has gained popularity as a replacement for numerical methods to reduce the computational budget. PINN uses physical information in the form of differential equations to enhance the performance of the neural networks. Though it works efficiently, the choice of the design of experiment scheme is important as the accuracy of the predicted responses using PINN depends on the training data. In this study, five different PDEs are used for numerical purposes, i.e., viscous Burger's equation, Shrödinger equation, heat equation, Allen-Cahn equation, and Korteweg-de Vries equation. A comparative study is performed to establish the necessity of the selection of a DoE scheme. It is seen that the Hammersley sampling-based PINN performs better than other DoE sample strategies.

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Section: Surrogate Modelmentioning

confidence: 99%

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Das¹,

Tesfamariam²

2022

Preprint

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“…whose results were compared with those obtained by the Lagrangian-Eulerian and Lax-Friedrichs schemes. While Patel et al (2022) proposes a PINN for discovering thermodynamically consistent equations that ensure hyperbolicity for inverse problems in shock hydrodynamics.…”

Section: Hyperbolic Equationsmentioning

confidence: 99%

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Cuomo¹,

Cola²,

Giampaolo³

et al. 2022

Preprint

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Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, and integral-differential equations. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages, the

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“…In the PINNs literature, Mao et al [3] first solved both forward and inverse one and two-dimensional Euler equations involving shocks in the Cartesian domain. Recently, Patel et al [21] introduced thermodynamically consistent physics-informed neural networks, which enforce the entropy conditions for the scalar conservation laws and the one-dimensional Euler equations. In [22] Fuks and Tchelepi proposed a stable way for PINNs to handle scalar conservation laws involving discontinuities.…”

Section: Introductionmentioning

confidence: 99%

Physics-informed neural networks for inverse problems in supersonic flows

Jagtap,

Mao,

Adams

et al. 2022

Preprint

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Accurate solutions to inverse supersonic compressible flow problems are often required for designing specialized aerospace vehicles. In particular, we consider the problem where we have data available for density gradients from Schlieren photography as well as data at the inflow and part of wall boundaries. These inverse problems are notoriously difficult and traditional methods may not be adequate to solve such ill-posed inverse problems. To this end, we employ the physics-informed neural networks (PINNs) and its extended version, extended PINNs (XPINNs), where domain decomposition allows to deploy locally powerful neural networks in each subdomain, which can provide additional expressivity in subdomains, where a complex solution is expected. Apart from the governing compressible Euler equations, we also enforce the entropy conditions in order to obtain viscosity solutions. Moreover, we enforce positivity conditions on density and pressure. We consider inverse problems involving two-dimensional expansion waves, two-dimensional oblique and bow shock waves. We compare solutions obtained by PINNs and XPINNs and invoke some theoretical results that can be used to decide on the generalization errors of the two methods.

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Thermodynamically consistent physics-informed neural networks for hyperbolic systems

Cited by 80 publications

References 57 publications

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning

Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next

Physics-informed neural networks for inverse problems in supersonic flows

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