Rotational and reflectional equivariant convolutional neural network for data-limited applications: Multiphase flow demonstration

Siddani, Bhargav Sriram; Balachandar, S.; Fang, Ruogu

doi:10.1063/5.0066049

Cited by 21 publications

(13 citation statements)

References 30 publications

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“…Another class of methods pertains to the customized neural network architectures that encode the prior physical or mathematical knowledge as hard constraints. Some of the examples of this methods applied in fluid dynamics are tensor basis neural network [42], transformation invariant neural network [25], physics-embedded neural network [43], spatial transformer [44,45], and equivariant networks [46,47].…”

Section: Introductionmentioning

confidence: 99%

Frame invariant neural network closures for Kraichnan turbulence

Pawar,

San,

Rasheed

et al. 2022

Preprint

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Numerical simulations of geophysical and atmospheric flows have to rely on parameterizations of subgrid scale processes due to their limited spatial resolution. Despite substantial progress in developing parameterization (or closure) models for subgrid scale (SGS) processes using physical insights and mathematical approximations, they remain imperfect and can lead to inaccurate predictions. In recent years, machine learning has been successful in extracting complex patterns from high-resolution spatio-temporal data, leading to improved parameterization models, and ultimately better coarse grid prediction. However, the inability to satisfy known physics and poor generalization hinders the application of these models for real-world problems. In this work, we propose a frame invariant closure approach to improve the accuracy and generalizability of deep learning-based subgrid scale closure models by embedding physical symmetries directly into the structure of the neural network. Specifically, we utilized specialized layers within the convolutional neural network in such a way that desired constraints are theoretically guaranteed without the need for any regularization terms. We demonstrate our framework for a two-dimensional decaying turbulence test case mostly characterized by the forward enstrophy cascade. We show that our frame invariant SGS model (i) accurately predicts the subgrid scale source term, (ii) respects the physical symmetries such as translation, Galilean, and rotation invariance, and (iii) is numerically stable when implemented in coarse-grid simulation with generalization to different initial conditions and Reynolds number. This work builds a bridge between extensive physics-based theories and data-driven modeling paradigms, and thus represents a promising step towards the development of physically consistent data-driven turbulence closure models.

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Section: Introductionmentioning

confidence: 99%

Frame invariant neural network closures for Kraichnan turbulence

Pawar,

San,

Rasheed

et al. 2022

Preprint

View full text Add to dashboard Cite

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“…The smaller number of parameters suggests that an equivariant model is trainable with a smaller size of data. 15 This suggestion was confirmed in Appendix D 2. Fig.…”

Section: Resultsmentioning

confidence: 54%

“…A smaller number of trainable parameters implies that an equivariant model is trainable with a smaller size of training data. 15 Fig. 19 shows the NERs against the various sizes of training data.…”

Section: Dependency On the Size Of Training Datamentioning

confidence: 99%

“…Such NNs incorporating geometric symmetries have started to be applied to various systems 11,12 including fluid systems. [13][14][15][16][17] Super-resolution (SR) refers to methods of estimating highresolution images from low-resolution ones. Super-resolution is studied in computer vision as an application of NNs.…”

Section: Introductionmentioning

confidence: 99%

“…They successfully predicted a vector quantity, namely the wall shear stress, on the surface of three-dimensional ar-teries. Siddai et al 15 developed an equivariant CNN and estimated three-dimensional steady flows around several particles from the particle positions, the mean flow, and the Reynolds number. Pawar et al 16 proposed a subgrid-scale model for the two-dimensional turbulence using an equivariant CNN.…”

Section: Introductionmentioning

confidence: 99%

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Roto-Translation Equivariant Super-Resolution of Two-Dimensional Flows Using Convolutional Neural Networks

Yasuda¹,

Onishi²

2022

Preprint

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Convolutional neural networks (CNNs) often process vectors as quantities having no direction like colors in images. This study investigates the effect of treating vectors as geometrical objects in terms of super-resolution of velocity on two-dimensional fluids. Vector is distinguished from scalar by the transformation law associated with a change in basis, which can be incorporated as the prior knowledge using the equivariant deep learning. We convert existing CNNs into equivariant ones by making each layer equivariant with respect to rotation and translation. The training data in the low-and high-resolution are generated with the downsampling or the spectral nudging. When the data inherit the rotational symmetry, the equivariant CNNs show comparable accuracy with the non-equivariant ones. Since the number of parameters is smaller in the equivariant CNNs, these models are trainable with a smaller size of the data. In this case, the transformation law of vector should be incorporated as the prior knowledge, where vector is explicitly treated as a quantity having direction. Two examples demonstrate that the symmetry of the data can be broken. In the first case, a downsampling method makes the correspondence between low-and high-resolution patterns dependent on the orientation. In the second case, the input data are insufficient to recognize the rotation of coordinates in the experiment with the spectral nudging. In both cases, the accuracy of the CNNs deteriorates if the equivariance is forced to be imposed, and the usage of conventional CNNs may be justified even though vector is processed as a quantity having no direction.

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Rotation invariance and equivariance in 3D deep learning: a survey

Fei,

Deng

2024

Artif Intell Rev

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Deep neural networks (DNNs) in 3D scenes show a strong capability of extracting high-level semantic features and significantly promote research in the 3D field. 3D shapes and scenes often exhibit complicated transformation symmetries, where rotation is a challenging and necessary subject. To this end, many rotation invariant and equivariant methods have been proposed. In this survey, we systematically organize and comprehensively overview all methods. First, we rewrite the previous definition of rotation invariance and equivariance by classifying them into weak and strong categories. Second, we provide a unified theoretical framework to analyze these methods, especially weak rotation invariant and equivariant ones that are seldom analyzed theoretically. We then divide existing methods into two main categories, i.e., rotation invariant ones and rotation equivariant ones, which are further subclassified in terms of manipulating input ways and basic equivariant block structures, respectively. In each subcategory, their common essence is highlighted, a couple of representative methods are analyzed, and insightful comments on their pros and cons are given. Furthermore, we deliver a general overview of relevant applications and datasets for two popular tasks of 3D semantic understanding and molecule-related. Finally, we provide several open problems and future research directions based on challenges and difficulties in ongoing research.

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Rotational and reflectional equivariant convolutional neural network for data-limited applications: Multiphase flow demonstration

Cited by 21 publications

References 30 publications

Frame invariant neural network closures for Kraichnan turbulence

Frame invariant neural network closures for Kraichnan turbulence

Roto-Translation Equivariant Super-Resolution of Two-Dimensional Flows Using Convolutional Neural Networks

Rotation invariance and equivariance in 3D deep learning: a survey

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