A technique for characterising feature size and quality of manifolds

Armstrong, E. F.; Sutherland, James C.

doi:10.1080/13647830.2021.1931715

Cited by 11 publications

(15 citation statements)

References 34 publications

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“…As recently proposed in Ref. 30 , we compute the normalized variance of a dependent variable, N (σ ) , for selected length scales, σ , on a manifold. We then analyze the derivative of N (σ ) with respect to σ , denoted as D(σ ) .…”

Section: Resultsmentioning

confidence: 99%

“…The detailed mathematical description of N (σ ) and D(σ ) can be found in the "Methods" section and in Ref. 30 . Given q manifold parameters, η η η = [η 1 , η 2 , .…”

Section: Resultsmentioning

confidence: 99%

“…A more in-depth discussion of how scales of variation can be linked to data density on a manifold grid can be found in Ref. 30 .…”

Section: Cost Function Response To Feature Size and Non-uniquenessmentioning

confidence: 99%

“…With noisy data, such as seen from experiment, it can be difficult to visually assess the quality of manifold topologies, especially in higher dimensions. The proposed cost function should adequately assess such topologies since the D(σ ) curves remain smooth even for noisy data 30 . Figure 6b demonstrates how the cost function behaves with increasing dimensionality of a projection using PCA, up to the original dimensionality of the system.…”

Section: The Different Projections Seen Inmentioning

confidence: 99%

“…In this paper, we are motivated by the emerging discussion on the need to characterize the quality of lowdimensional manifolds [26][27][28][29][30] . Quantitative tools are needed in areas where researchers tune the hyper-parameters of dimensionality reduction or manifold learning techniques to obtain improved manifolds of particular types of data [31][32][33][34][35] .…”

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confidence: 99%

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Cost function for low-dimensional manifold topology assessment

Zdybał

Armstrong

Sutherland

et al. 2022

Sci Rep

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In reduced-order modeling, complex systems that exhibit high state-space dimensionality are described and evolved using a small number of parameters. These parameters can be obtained in a data-driven way, where a high-dimensional dataset is projected onto a lower-dimensional basis. A complex system is then restricted to states on a low-dimensional manifold where it can be efficiently modeled. While this approach brings computational benefits, obtaining a good quality of the manifold topology becomes a crucial aspect when models, such as nonlinear regression, are built on top of the manifold. Here, we present a quantitative metric for characterizing manifold topologies. Our metric pays attention to non-uniqueness and spatial gradients in physical quantities of interest, and can be applied to manifolds of arbitrary dimensionality. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional projections can be found. We delineate a few applications of the cost function to datasets representing argon plasma, reacting flows and atmospheric pollutant dispersion. We demonstrate how the cost function can assess various dimensionality reduction and manifold learning techniques as well as data preprocessing strategies in their capacity to yield quality low-dimensional projections. We show that improved manifold topologies can facilitate building nonlinear regression models.

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Section: Resultsmentioning

confidence: 99%

“…The detailed mathematical description of N (σ ) and D(σ ) can be found in the "Methods" section and in Ref. 30 . Given q manifold parameters, η η η = [η 1 , η 2 , .…”

Section: Resultsmentioning

confidence: 99%

“…A more in-depth discussion of how scales of variation can be linked to data density on a manifold grid can be found in Ref. 30 .…”

Section: Cost Function Response To Feature Size and Non-uniquenessmentioning

confidence: 99%

Section: The Different Projections Seen Inmentioning

confidence: 99%

mentioning

confidence: 99%

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Cost function for low-dimensional manifold topology assessment

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Armstrong

Sutherland

et al. 2022

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Reduced-Order Modeling of Reacting Flows Using Data-Driven Approaches

Zdybał

Malik

Coussement

et al. 2023

Lecture Notes in Energy

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Data-driven modeling of complex dynamical systems is becoming increasingly popular across various domains of science and engineering. This is thanks to advances in numerical computing, which provides high fidelity data, and to algorithm development in data science and machine learning. Simulations of multicomponent reacting flows can particularly profit from data-based reduced-order modeling (ROM). The original system of coupled partial differential equations that describes a reacting flow is often large due to high number of chemical species involved. While the datasets from reacting flow simulation have high state-space dimensionality, they also exhibit attracting low-dimensional manifolds (LDMs). Data-driven approaches can be used to obtain and parameterize these LDMs. Evolving the reacting system using a smaller number of parameters can yield substantial model reduction and savings in computational cost. In this chapter, we review recent advances in ROM of turbulent reacting flows. We demonstrate the entire ROM workflow with a particular focus on obtaining the training datasets and data science and machine learning techniques such as dimensionality reduction and nonlinear regression. We present recent results from ROM-based simulations of experimentally measured Sandia flames D and F. We also delineate a few remaining challenges and possible future directions to address them. This chapter is accompanied by illustrative examples using the recently developed Python software, PCAfold. The software can be used to obtain, analyze and improve low-dimensional data representations. The examples provided herein can be helpful to students and researchers learning to apply dimensionality reduction, manifold approaches and nonlinear regression to their problems. The Jupyter notebook with the examples shown in this chapter can be found on GitHub at https://github.com/kamilazdybal/ROM-of-reacting-flows-Springer.

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SVD perspectives for augmenting DeepONet flexibility and interpretability

Venturi

Casey²

2023

Computer Methods in Applied Mechanics and Engineering

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A technique for characterising feature size and quality of manifolds

Cited by 11 publications

References 34 publications

Cost function for low-dimensional manifold topology assessment

Cost function for low-dimensional manifold topology assessment

Reduced-Order Modeling of Reacting Flows Using Data-Driven Approaches

SVD perspectives for augmenting DeepONet flexibility and interpretability

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