A Physics-Informed Neural Network for Quantifying the Microstructural Properties of Polycrystalline Nickel Using Ultrasound Data: A promising approach for solving inverse problems

Shukla, K. K.; Jagtap, Ameya D.; Blackshire, James L.; Sparkman, Daniel; Karniadakis, George Em

doi:10.1109/msp.2021.3118904

Cited by 74 publications

(17 citation statements)

References 16 publications

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“…Due to such sparse, noisy data or incomplete knowledge of the underlying physical laws the traditional methods are not appropriate, but the PINN can be easily employed. In the literature, various inverse problems are solved by PINN, see for example, [34,13,35,15,11,10,16,36].…”

Section: Physics-informed Neural Networkmentioning

confidence: 99%

“…This was first addressed by domain decomposition based PINN methodology namely, conservative PINN (cPINN) methodology, see Jagtap et al [10], and further by more general space-time domain decomposition based extended PINN (XPINN) methodology, see Jagtap & Karniadakis [11], and for the theory of XPINN see [12]. PINNs have been successfully applied to solve many problems in the field of computational science, see [13,14,15,16,17,18] for more details. In a recent study [19] establishes the mathematical foundation of the PINNs, whereas [20] presented estimates on the generalization error of the PINN methodology.…”

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confidence: 99%

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Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations

Jagtap¹,

Mitsotakis²,

Karniadakis³

2022

Preprint

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A. We consider strongly-nonlinear and weakly-dispersive surface water waves governed by equations of Boussinesq type, known as the Serre-Green-Naghdi system; it describes future states of the free water surface and depth averaged horizontal velocity, given their initial state. The lack of knowledge of the velocity field as well as the initial states provided by measurements lead to an ill-posed problem that cannot be solved by traditional techniques. To this end, we employ physics-informed neural networks (PINNs) to generate solutions to such ill-posed problems using only data of the free surface elevation and depth of the water. PINNs can readily incorporate the physical laws and the observational data, thereby enabling inference of the physical quantities of interest. In the present study, both experimental and synthetic (generated by numerical methods) training data are used to train PINNs. Furthermore, multi-fidelity data are used to solve the inverse water wave problem by leveraging both high-and low-fidelity data sets. The applicability of the PINN methodology for the estimation of the impact of water waves onto solid obstacles is demonstrated after deriving the corresponding equations. The present methodology can be employed to efficiently design offshore structures such as oil platforms, wind turbines, etc. by solving the corresponding ill-posed inverse water waves problem.

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Section: Physics-informed Neural Networkmentioning

confidence: 99%

mentioning

confidence: 99%

Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations

Jagtap¹,

Mitsotakis²,

Karniadakis³

2022

Preprint

Self Cite

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“…Since then, there has been an explosive growth in designing and applying PINNs for a variety of applications involving PDEs. A very incomplete list of references includes [36,28,33,45,12,13,14,16,29,30,31,2,40,15,11,41] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for physics informed neural networks approximating the Navier-Stokes equations

Ryck¹,

Jagtap²,

Mishra³

2022

Preprint

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We prove rigorous bounds on the errors resulting from the approximation of the incompressible Navier-Stokes equations with (extended) physics informed neural networks. We show that the underlying PDE residual can be made arbitrarily small for tanh neural networks with two hidden layers. Moreover, the total error can be estimated in terms of the training error, network size and number of quadrature points. The theory is illustrated with numerical experiments.

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“…In recent years physics-informed neural networks (PINNs) [1] emerged as an alternative simple method to solve many problems in computational science and engineering, see, for example [2,3,4,5,6,7,8,9,10,4,11,12,13,14,15]. In particular, PINNs do not require meshes and can efficiently solve forward problems and even ill-posed inverse problems, which are otherwise difficult or sometimes even impossible to solve using traditional numerical methods.…”

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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A Physics-Informed Neural Network for Quantifying the Microstructural Properties of Polycrystalline Nickel Using Ultrasound Data: A promising approach for solving inverse problems

Cited by 74 publications

References 16 publications

Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations

Deep learning of inverse water waves problems using multi-fidelity data: Application to Serre-Green-Naghdi equations

Error estimates for physics informed neural networks approximating the Navier-Stokes equations

Physics-informed neural networks for inverse problems in supersonic flows

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