Constraint-aware neural networks for Riemann problems

Magiera, Jim; Ray, Deep; Hesthaven, Jan S.; Rohde, Christian

doi:10.1016/j.jcp.2020.109345

Cited by 54 publications

(26 citation statements)

References 36 publications

(95 reference statements)

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“…This is the approach followed by several authors in the context of physical simulations, which aim to solve a set of partial differential equations (PDEs) in complex dynamical systems. Physical problems must satisfy inherently certain conditions dictated by physics, often formulated as conservation laws, and can be imposed to a neural network using extra loss terms in the constrained optimization process [32].…”

Section: Introductionmentioning

confidence: 99%

Structure-preserving neural networks

Hernández

Badías²,

González³

et al. 2021

Journal of Computational Physics

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We develop a method to learn physical systems from data that employs feedforward neural networks and whose predictions comply with the first and second principles of thermodynamics. The method employs a minimum amount of data by enforcing the metriplectic structure of dissipative Hamiltonian systems in the form of the socalled General Equation for the Non-Equilibrium Reversible-Irreversible Coupling, GENERIC (Öttinger and Grmela (1997) [36]). The method does not need to enforce any kind of balance equation, and thus no previous knowledge on the nature of the system is needed. Conservation of energy and dissipation of entropy in the prediction of previously unseen situations arise as a natural by-product of the structure of the method. Examples of the performance of the method are shown that comprise conservative as well as dissipative systems, discrete as well as continuous ones.

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Section: Introductionmentioning

confidence: 99%

Structure-preserving neural networks

Hernández

Badías²,

González³

et al. 2021

Journal of Computational Physics

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“…The overview paper by Karpatne et al (2017) provides a taxonomy for theory-guided data science, with the goal of incorporating scientific consistency in the learning of generalizable models. Much research in physics-informed machine learning has focused on incorporating constraints in neural networks (Ling et al, 2016;Jones et al, 2018), often through the use of objective/loss functions that penalize constraint violation (Magiera et al, 2020).…”

Section: Introductionmentioning

confidence: 99%

Tensor Basis Gaussian Process Models of Hyperelastic Materials

Frankel

Jones

Swiler

2020

J Mach Learn Model Comput

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In this work, we develop Gaussian process regression (GPR) models of isotropic hyperelastic material behavior. First, we consider the direct approach of modeling the components of the Cauchy stress tensor as a function of the components of the Finger stretch tensor in a Gaussian process. We then consider an improvement on this approach that embeds rotational invariance of the stress-stretch constitutive relation in the GPR representation. This approach requires fewer training examples and achieves higher accuracy while maintaining invariance to rotations exactly. Finally, we consider an approach that recovers the strain-energy density function and derives the stress tensor from this potential. Although the error of this model for predicting the stress tensor is higher, the strain-energy density is recovered with high accuracy from limited training data. The approaches presented here are examples of physics-informed machine learning. They go beyond purely data-driven approaches by embedding the physical system constraints directly into the Gaussian process representation of materials models.

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“…Several studies have been conducted to model the dynamics of chaotic fluid flow using ML algorithms [25][26][27][28][29][30][31]. Recently there is a growing interest in using the physics of the problem in combination with the data-driven algorithms [28,[32][33][34][35][36][37][38]. The physics can be incorporated into these learning algorithms by adding a regularization term (based on governing equations) in loss function or modifying the neural network architecture to enforce certain physical constraints.In addition to reduced order modeling and chaotic dynamical systems, the turbulence closure problem has also benefited from the application of ML algorithms and has led to reducing uncertainties in .…”

mentioning

confidence: 99%

A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

Pawar

San

Rasheed

et al. 2020

Theor. Comput. Fluid Dyn.

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In the present study, we investigate different data-driven parameterizations for large eddy simulation of two-dimensional turbulence in the a priori settings. These models utilize resolved flow field variables on the coarser grid to estimate the subgrid-scale stresses. We use data-driven closure models based on localized learning that employs multilayer feedforward artificial neural network (ANN) with point-to-point mapping and neighboring stencil data mapping, and convolutional neural network (CNN) fed by data snapshots of the whole domain. The performance of these data-driven closure models is measured through a probability density function and is compared with the dynamic Smagorinksy model (DSM). The quantitative performance is evaluated using the cross-correlation coefficient between the true and predicted stresses. We analyze different frameworks in terms of the amount of training data, selection of input and output features, their characteristics in modeling with accuracy, and training and deployment computational time. We also demonstrate computational gain that can be achieved using the intelligent eddy viscosity model that learns eddy viscosity computed by the DSM instead of subgrid-scale stresses. We detail the hyperparameters optimization of these models using the grid search algorithm.Keywords Turbulence closure · Deep learning · Neural networks · Subgrid scale modeling · Large eddy simulation 1 Introduction Direct numerical simulation (DNS) of complex fluid flows encountered in many engineering and geophysical applications is computationally unmanageable because of the need to resolve a wide range of spatiotemporal scales. Large eddy simulation (LES) and Reynolds Averaged Navier-Stokes (RANS) modeling are two most commonly used mathematical modeling frameworks that give accurate predictions by considering the interaction between the unresolved and grid-resolved scales. The development of these models is termed as the turbulence closure problem and has been a long-standing challenge in fluid mechanics community [1][2][3][4].In LES, we filter the Navier-Stokes equations using a low-pass filtering operator that separates the motion into small and large scales, and in turn, produces modified equations which are computationally faster to solve than actual Navier-Stokes equations [5][6][7]. The interaction between grid-resolved and unresolved scales is then taken into account by introducing subgrid-scale stress (SGS) term in the modified equation. The main task of the SGS model is to provide mean dissipation that corresponds to the transfer of energy from resolved scales to unresolved scales (the production of energy at large scales is balanced by the dissipation of energy at small scales based on Kolmogorov's theory of turbulence). The dissipation effect of unresolved scales can be included utilizing an eddy viscosity parameterization obtained through grid-resolved quantities. These approaches are called as functional approaches which assume isotropy of small scales to present the average dissipation of t...

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Constraint-aware neural networks for Riemann problems

Cited by 54 publications

References 36 publications

Structure-preserving neural networks

Structure-preserving neural networks

Tensor Basis Gaussian Process Models of Hyperelastic Materials

A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

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