Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport

Lu, Lu; Pestourie, Raphaël; Johnson, Steven G.; Romano, Giuseppe

doi:10.48550/arxiv.2204.06684

Cited by 6 publications

(7 citation statements)

References 48 publications

(65 reference statements)

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“…-To reduce the dependency of large paired datasets (when accurate information about the governing law of the physical system is not available), Deep Transfer Operator Learning (DeepONet) [8] can be used for accurate prediction of quantities of interest in related domains, with only a handful of new measurements. -Develop faster ways to train neural operators by incorporating multimodality and multifidelity data [55,56,57]. -The application of neural operators in life sciences is endless.…”

Section: Discussionmentioning

confidence: 99%

“…To that end, training a DeepONet or any other neural operator with only high fidelity data would be computationally expensive. One promising way to solve such realistic problems is using multifidelity approaches proposed in [57,55,56]. These real problems take leverage of the generalized flavor of DeepONet, which can be flexibly designed for any problem at hand.…”

Section: Layout 2: 64 Turbinesmentioning

confidence: 99%

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Physics-Informed Deep Neural Operator Networks

Goswami¹,

Bora²,

Yu³

et al. 2022

Preprint

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Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if the governing equations are known they can be incorporated into the loss function during training to develop physics-informed neural operators. Neural operators can be used as surrogates in design problems, uncertainty quantification, autonomous systems, and almost in any application requiring real-time inference. Moreover, independently pre-trained DeepONets can be used as components of a complex multi-physics system by coupling them together with relatively light training. Here, we present a review of DeepONet, the Fourier neural operator, and the graph neural operator, as well as appropriate extensions with feature expansions, and highlight their usefulness in diverse applications in computational mechanics, including porous media, fluid mechanics, and solid mechanics.

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Section: Discussionmentioning

confidence: 99%

Section: Layout 2: 64 Turbinesmentioning

confidence: 99%

Physics-Informed Deep Neural Operator Networks

Goswami¹,

Bora²,

Yu³

et al. 2022

Preprint

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“…DeepXDE also features NNs for nonlinear operator learning such as DeepONet [97], POD-DeepONet [98], MIONet [99], DeepM&Mnet [100,101] and multifidelity DeepONet [102].…”

Section: Deepxdementioning

confidence: 99%

Stiff-PDEs and Physics-Informed Neural Networks

Sharma

Evans

Tindall

et al. 2023

Arch Computat Methods Eng

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In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.

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“…However, to train DNNs, solutions of PDEs are required as a training dataset, which must be analytically or numerically provided in advance. The deep operator network (DeepONet, [18]) has been also robustly studied lately in multiple fields, such as multiphysics and multiscale problems in [50], [51], a method using Fourier Neural Operator in [52], a method for multiple-input operators in [53], and a method for input data with various degrees of accuracy in [54]. The authors in [20], [21] introduced an operator learning approach that can learn multiple PDE solutions.…”

Section: Related Workmentioning

confidence: 99%

Unsupervised Legendre–Galerkin Neural Network for Solving Partial Differential Equations

2023

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In recent years, machine learning methods have been used to solve partial differential equations (PDEs) and dynamical systems, leading to the development of a new research field called scientific machine learning, which combines techniques such as deep neural networks and statistical learning with classical problems in applied mathematics. In this paper, we present a novel numerical algorithm that uses machine learning and artificial intelligence to solve PDEs. Based on the Legendre-Galerkin framework, we propose an unsupervised machine learning algorithm that learns multiple instances of the solutions for different types of PDEs. Our approach addresses the limitations of both data-driven and physics-based methods. We apply the proposed neural network to general 1D and 2D PDEs with various boundary conditions, as well as convection-dominated singularly perturbed PDEs that exhibit strong boundary layer behavior.

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Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport

Cited by 6 publications

References 48 publications

Physics-Informed Deep Neural Operator Networks

Physics-Informed Deep Neural Operator Networks

Stiff-PDEs and Physics-Informed Neural Networks

Unsupervised Legendre–Galerkin Neural Network for Solving Partial Differential Equations

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