Computational multiscale methods for quasi-gas dynamic equations

Четверушкин, Б. Н.; Chung, Eric T.; Efendiev, Yalchin; Pun, Sai-Mang; Zhang, Zecheng

doi:10.1016/j.jcp.2021.110352

Cited by 18 publications

(4 citation statements)

References 39 publications

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“…This is referred to as free-BelNet. As a benchmark, we also refer to established methods [43][44][45][46][47] to solve multiscale parametric PDEs.…”

Section: Numerical Examplesmentioning

confidence: 99%

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BelNet: basis enhanced learning, a mesh-free neural operator

Zhang,

Wing Tat,

Schaeffer

2023

Proc. R. Soc. A.

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Operator learning trains a neural network to map functions to functions. An ideal operator learning framework should be mesh-free in the sense that the training does not require a particular choice of discretization for the input functions, allows for the input and output functions to be on different domains, and is able to have different grids between samples. We propose a mesh-free neural operator for solving parametric partial differential equations. The basis enhanced learning network (BelNet) projects the input function into a latent space and reconstructs the output functions. In particular, we construct part of the network to learn the ‘basis’ functions in the training process. This generalized the networks proposed in Chen & Chen (Chen and Chen 1995 IEEE Trans. Neural Netw. 49 , 911–917. ( doi:10.1109/72.392253 ) and 6 , 904–910. ( doi:10.1109/IJCNN.1993.716815 )) to account for differences in input and output meshes. Through several challenging high-contrast and multiscale problems, we show that our approach outperforms other operator learning methods for these tasks and allows for more freedom in the sampling and/or discretization process.

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“…This is referred to as free-BelNet. As a benchmark, we also refer to established methods [43][44][45][46][47] to solve multiscale parametric PDEs.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…We compare our results to DON and the solution generated by the generalized multiscale finite-element method (GMsFEM) [45,46]. For GMsFEM, the fine mesh size is set to 50×50, the coarse element size is 2×2, and we use the first three GMsFEM basis.…”

Section: Numerical Examplesmentioning

confidence: 99%

BelNet: basis enhanced learning, a mesh-free neural operator

Zhang,

Wing Tat,

Schaeffer

2023

Proc. R. Soc. A.

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“…1 Input: current x = (x 1 , x 2 ) 2 Initialize: set the size related parameters r 1 , r 2 > 0 and speed related parameter s 0 , and draw a random number s ∼ U(0, s t ). We then define l 1 = s 0 + s + r 1 and l 2 = s 0 + s + r 2 3 Create a rectangle: the top right vertex coordinate is (x 1 + l 1 Δx, x 2 + l 2 Δx) and the size is equal to (2r 1 Δx, 2r 2 Δx) 4 Draw a point x from the rectangle above uniformly; this point will be the next sample The second permeability field κ 2 (x) (see figure 2 right for the illustration) has multiple high contrast channels and is widely used in the multiscale finite element method society [4,6].…”

Section: Permeability Fieldsmentioning

confidence: 99%

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

Lin¹,

Zhang²,

Zhang³

2022

Inverse Problems

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We consider the inverse source problem in the parabolic equation, where the unknown source possesses the semi-discrete formulation. Theoretically, we prove that the ﬂux data from any nonempty open subset of the boundary can uniquely determine the semi-discrete source. This means the observed area can be extremely small, and that is why we call the data as sparse boundary data. For the numerical reconstruction, we formulate the problem from the Bayesian sequential prediction perspective and conduct the numerical examples which estimate the space-time-dependent source state by state. To better demonstrate the performance of the method, we solve two common multiscale problems from two models with a long sequence of the source. The numerical results illustrate that the inversion is accurate and eﬃcient.

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“…Multiscale methods have extensively been studied in the literature. For linear problems, homogenization-based approaches [23,39,40], multiscale finite element methods [23,31,35], generalized multiscale finite element methods (GMsFEM) [12,13,14,17,21], Constraint Energy Minimizing GMsFEM (CEM-GMsFEM) [15,16,7], nonlocal multi-continua (NLMC) approaches [19], metric-based upscaling [44], heterogeneous multiscale method [20], localized orthogonal decomposition (LOD) [30], equation-free approaches [45,46], multiscale stochastic approaches [33,34,32], and hierarchical multiscale methods [5] have been studied. Approaches such as GMsFEM and NLMC are designed to handle high-contrast problems.…”

Section: Introductionmentioning

confidence: 99%

Hybrid explicit-implicit learning for multiscale problems with time dependent source

Efendiev¹,

Leung²,

Li³

et al. 2022

Preprint

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The splitting method is a powerful method for solving partial differential equations. Various splitting methods have been designed to separate different physics, nonlinearities, and so on. Recently, a new splitting approach has been proposed where some degrees of freedom are handled implicitly while other degrees of freedom are handled explicitly. As a result, the scheme contains two equations, one implicit and the other explicit. The stability of this approach has been studied. It was shown that the time step scales as the coarse spatial mesh size, which can provide a significant computational advantage. However, the implicit solution part can be expensive, especially for nonlinear problems. In this paper, we introduce modified partial machine learning algorithms to replace the implicit solution part of the algorithm. These algorithms are first introduced in 'HEI: Hybrid Explicit-Implicit Learning For Multiscale Problems', where a homogeneous source term is considered along with the Transformer, which is a neural network that can predict future dynamics. In this paper, we consider time-dependent source terms which is a generalization of the previous work. Moreover, we use the whole history of the solution to train the network. As the implicit part of the equations is more complicated to solve, we design a neural network to predict it based on training. Furthermore, we compute the explicit part of the solution using our splitting strategy. In addition, we use Proper Orthogonal Decomposition based model reduction in machine learning. The machine learning algorithms provide computational saving without sacrificing accuracy. We present three numerical examples which show that our machine learning scheme is stable and accurate.

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Computational multiscale methods for quasi-gas dynamic equations

Cited by 18 publications

References 39 publications

BelNet: basis enhanced learning, a mesh-free neural operator

BelNet: basis enhanced learning, a mesh-free neural operator

Theoretical and numerical studies of inverse source problem for the linear parabolic equation with sparse boundary measurements

Hybrid explicit-implicit learning for multiscale problems with time dependent source

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