Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

Kneer, Simon; Sayadi, Taraneh; Sipp, Denis; Schmid, Peter; Rigas, Georgios

doi:10.48550/arxiv.2111.02893

Cited by 5 publications

(5 citation statements)

References 25 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

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“…In the physics-informed DMD developed by Baddoo et al (2021), the DMD matrix that provides the best-fit linear system to the data is constrained to the space of matrices that commute with the symmetry operators. Finally, Kneer et al (2022) utilize the so-called spatial translation networks to perform template matching akin to that of Rowley & Marsden (2000). Each of these methods comes with a new set of technical difficulties, and it is unclear whether they are practical for the three-dimensional complex fluid flows that we consider here.…”

Section: Symmetries and Symmetry Reductionmentioning

confidence: 99%

Symmetry-reduced dynamic mode decomposition of near-wall turbulence

et al. 2022

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Data-driven dimensionality reduction methods such as proper orthogonal decomposition and dynamic mode decomposition have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration, as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.

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“…This is done by applying the discrete operation that rotates and shiftreflects the vorticity snapshots and selecting the snapshot that minimizes the norm. We note that we take a different approach for reducing the symmetries compared to previous research on symmetry-aware AEs [34].…”

Section: Data-driven Dimension Reduction and Dynamic Modeling A Dimen...mentioning

confidence: 99%

Data-driven low-dimensional dynamic model of Kolmogorov flow

Jesús¹,

Graham²

2022

Preprint

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Reduced order models (ROMs) that capture flow dynamics are of interest for decreasing computational costs for simulation as well as for model-based control approaches. This work presents a data-driven framework for minimal-dimensional models that effectively capture the dynamics and properties of the flow. We apply this to Kolmogorov flow in a regime consisting of chaotic and intermittent behavior, which is common in many flows processes and is challenging to model. The trajectory of the flow travels near relative periodic orbits (RPOs), interspersed with sporadic bursting events corresponding to excursions between the regions containing the RPOs. The first step in development of the models is use of an undercomplete autoencoder to map from the full state data down to a latent space of dramatically lower dimension. Then models of the discrete-time evolution of the dynamics in the latent space are developed. By analyzing the model performance as a function of latent space dimension we can estimate the minimum number of dimensions required to capture the system dynamics. To further reduce the dimension of the dynamical model, we factor out a phase variable in the direction of translational invariance for the flow, leading to separate evolution equations for the pattern and phase dynamics. At a model dimension of five for the pattern dynamics, as opposed to the full state dimension of 1024 (i.e. a 32 × 32 grid), accurate predictions are found for individual trajectories out to about two Lyapunov times, as well as for long-time statistics. The nearly heteroclinic connections between the different RPOs, including the quiescent and bursting time scales, are well captured. We also capture key features of the phase dynamics. Finally, we use the low-dimensional representation to predict future bursting events, finding good success.

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Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

Cited by 5 publications

References 25 publications

Frame invariant neural network closures for Kraichnan turbulence

Frame invariant neural network closures for Kraichnan turbulence

Symmetry-reduced dynamic mode decomposition of near-wall turbulence

Data-driven low-dimensional dynamic model of Kolmogorov flow

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