Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows

Wenzel, Tizian; Kurz, Marius; Beck, Andrea; Santin, Gabriele; Haasdonk, Bernard

doi:10.48550/arxiv.2103.13655

Cited by 2 publications

(2 citation statements)

References 7 publications

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“…Subsequently, a classical PCA is applied to identify the low-dimensional space. Additional methods to construct such a mapping are, e.g., the Gaussian process latent variable model (GPLVM) [35], self-organizing maps [32], diffeomorphic dimensionality reduction [56] and deep kernel networks [58].…”

Section: Nonlinear Symplectic Trial Manifold Based On Deep Convolutio...mentioning

confidence: 99%

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Buchfink¹,

Glas²,

Haasdonk³

2021

Preprint

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Classical model reduction techniques project the governing equations onto linear subspaces of the high-dimensional state-space. For problems with slowly decaying Kolmogorov-nwidths such as certain transport-dominated problems, however, classical linear-subspace reducedorder models (ROMs) of low dimension might yield inaccurate results. Thus, the concept of classical linear-subspace ROMs has to be extended to more general concepts, like Model Order Reduction (MOR) on manifolds. Moreover, as we are dealing with Hamiltonian systems, it is crucial that the underlying symplectic structure is preserved in the reduced model, as otherwise it could become unphysical in the sense that the energy is not conserved or stability properties are lost. To the best of our knowledge, existing literature addresses either MOR on manifolds or symplectic model reduction for Hamiltonian systems, but not their combination. In this work, we bridge the two aforementioned approaches by providing a novel projection technique called symplectic manifold Galerkin (SMG), which projects the Hamiltonian system onto a nonlinear symplectic trial manifold such that the reduced model is again a Hamiltonian system. We derive analytical results such as stability, energy-preservation and a rigorous a-posteriori error bound. Moreover, we construct a weakly symplectic deep convolutional autoencoder as a computationally practical approach to approximate a nonlinear symplectic trial manifold. Finally, we numerically demonstrate the ability of the method to outperform (non-)structure-preserving linear-subspace ROMs and non-structure-preserving MOR on manifold techniques.

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Section: Nonlinear Symplectic Trial Manifold Based On Deep Convolutio...mentioning

confidence: 99%

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Buchfink¹,

Glas²,

Haasdonk³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…To keep the presentation easy to follow, we do not include experimental results with those SDKN in the present article. But we refer to [37] for an example of a successful application and comparison of the SDKN to NN in the context of closure term prediction for computational fluid dynamics. We also have positive experience with the SDKN architecture in various other application settings such as spine modelling and frameworks such as kernel autoencoders which are subject of ongoing work.…”

Section: Introductionmentioning

confidence: 99%

Universality and Optimality of Structured Deep Kernel Networks

Wenzel¹,

Santin²,

Haasdonk³

2021

Preprint

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Kernel based methods yield approximation models that are flexible, efficient and powerful. In particular, they utilize fixed feature maps of the data, being often associated to strong analytical results that prove their accuracy. On the other hand, the recent success of machine learning methods has been driven by deep neural networks (NNs). They achieve a significant accuracy on very high-dimensional data, in that they are able to learn also efficient data representations or data-based feature maps.In this paper, we leverage a recent deep kernel representer theorem to connect the two approaches and understand their interplay. In particular, we show that the use of special types of kernels yield models reminiscent of neural networks that are founded in the same theoretical framework of classical kernel methods, while enjoying many computational properties of deep neural networks. Especially the introduced Structured Deep Kernel Networks (SDKNs) can be viewed as neural networks with optimizable activation functions obeying a representer theorem. Analytic properties show their universal approximation properties in different asymptotic regimes of unbounded number of centers, width and depth. Especially in the case of unbounded depth, the constructions is asymptotically better than corresponding constructions for ReLU neural networks, which is made possible by the flexibility of kernel approximation.

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Structured Deep Kernel Networks for Data-Driven Closure Terms of Turbulent Flows

Cited by 2 publications

References 7 publications

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Symplectic Model Reduction of Hamiltonian Systems on Nonlinear Manifolds

Universality and Optimality of Structured Deep Kernel Networks

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