Neural network enhanced computations on coarse grids

Nordström, Jan; Ålund, Oskar

doi:10.1016/j.jcp.2020.109821

Cited by 10 publications

(9 citation statements)

References 17 publications

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“…This procedure was developed in [31], where it was used to increase the convergence rate to steady state. It has also been used on linear problems to control error growth [34] and to aid coarsegrid computations [32]. In Article IV, we use the MPT technique together with data from the wall model to improve the coarse-grid results for the nonlinear INS equations.…”

Section: Article IVmentioning

confidence: 99%

Summation-by-parts formulations for flow problems

Laurén

2022

Linköping Studies in Science and Technology. Dissertations

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Many problems in engineering and physics can be described by partial differential equations (PDEs). Augmented with proper initial and boundary conditions, the PDE forms an initial-boundary value problem (IBVP). An IBVP is said to be well-posed if a unique solution bounded by the given data exists. The behavior of the solution as well as the well-posedness of the IBVP are heavily influenced by the form and position of the boundary conditions. The solution, if it exists, to an IBVP cannot in general be obtained in closed form. Instead, one can compute an approximation using numerical methods. Summation-by-parts (SBP) operators together with the simultaneous approximation term (SAT) technique can be used to form stable numerical methods that generate accurate approximations. All discretizations in this thesis are based on the SBP-SAT framework.The first part of the thesis concentrates on IBVPs describing fluid motion. Different sets of boundary conditions are investigated in terms of energy-boundedness and spectral properties. Wall models are also studied and used to aid coarse-grid simulations in an energy stable manner.In the second part of the thesis, special discretization operators are derived. First, single-block SBP operators are combined with interpolation operators. The result is a single SBP operators on a multi-block grid that encapsulates both the metric terms and the interface treatments. Second, new SBP operators are developed for an application in high-energy physics. The new operators preserves the unit-trace, which is important since it enables a probability interpretation of the density matrix.

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Section: Article IVmentioning

confidence: 99%

Summation-by-parts formulations for flow problems

Laurén

2022

Linköping Studies in Science and Technology. Dissertations

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“…In Paper I, we showed that for linear problems it was not sufficient to apply only dissipative boundary conditions along the physical and subdomain boundaries. To guarantee bounded energy for linear problems we introduced linear penalty terms [17,18] to the conservation laws in each domain to enforce two-way coupling.…”

Section: The Overset Domain Problemmentioning

confidence: 99%

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems II: Entropy Bounded Formulations for Nonlinear Conservation Laws

Kopriva¹,

Gassner²,

Nordström³

2022

Preprint

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We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. Entropy dissipation and additional coupling in the overlap region are added through the use of linear penalties.

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“…Deep neural networks are a system of interconnected computational nodes loosely based on biological neural networks and, mathematically, can be formulated as compositional functions [7,32]. In contrast to shallow neural networks, which are networks with just a single hidden layer, these NNs are composed of two or more hidden layers [3].…”

Section: Deep Neural Networkmentioning

confidence: 99%

“…Autodiff is a technique that is used in PINNs to compute the partial derivatives of the NN approximations and thus embed the governing PDEs and associated boundary conditions in the loss function. Given that it facilitates "mesh-less" numerical computations of derivatives, it endows PINNs with several advantages over traditional numerical discretisation approaches for solving PDEs (such as the finite difference and finite element methods) that can be computationally expensive due to complex mesh-generation [7,32,40]. For example, Refs.…”

Section: Physics-informed Neural Networkmentioning

confidence: 99%

“…In recent years there has been an increased interest in the use of neural networks (NNs) as functional approximators [1][2][3][4]. The interest lies in the fact that NNs are versatile as demonstrated in their success in various applications such as natural language processing, image recognition and, more recently, scientific computing [5][6][7]. In scientific computing, they have been shown to be robust and data-efficient solvers of partial differential equations that govern diverse systems studied in mathematics, science and engineering [5].…”

Section: Introductionmentioning

confidence: 99%

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Determining QNMs using PINNs

Cornell¹,

Ncube²,

Harmsen³

2022

Preprint

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In recent years there has been an increased interest in neural networks, particularly with regard to their ability to approximate partial differential equations. In this regard, research has begun on so-called physics-informed neural networks (PINNs) which incorporate into their loss function the boundary conditions of the functions they are attempting to approximate. In this paper, we investigate the viability of obtaining the quasi-normal modes (QNMs) of non-rotating black holes in 4-dimensional space-time using PINNs, and we find that it is achievable using a standard approach that is capable of solving eigenvalue problems (dubbed the eigenvalue solver here). In comparison to the QNMs obtained via more established methods (namely, the continued fraction method and the 6th-order Wentzel, Kramer, Brillouin method) the PINN computations share the same degree of accuracy as these counterparts. In other words, our PINN approximations had percentage deviations as low as (δω Re , δω Im ) = (< 0.01%, < 0.01%). In terms of efficiency, however, the PINN approach falls short, leading to our conclusion that the method is currently not to be recommended when considering overall performance.

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Neural network enhanced computations on coarse grids

Cited by 10 publications

References 17 publications

Summation-by-parts formulations for flow problems

Summation-by-parts formulations for flow problems

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems II: Entropy Bounded Formulations for Nonlinear Conservation Laws

Determining QNMs using PINNs

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