Frame invariant neural network closures for Kraichnan turbulence

Abstract

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.

Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES

“…A priori and a posteriori tests show that all these physics-constrained CNNs outperform the physics-agnostic CNN in the smalldata regime (n tr = 50). Improvements of the data-driven SGS closures using the GCNN, which uses an equivariant-preserving architecture, are consistent with the findings of a recent study [69], though it should be mentioned that DA, which simply builds equivariance in the training samples and can be used with any architecture, shows comparable or even in some cases better performance than GCNN. The spectra from the LES with physics-constrained CNNs better match the FDNS spectra especially at the tails (high-wavenumber structures).…”

“…While translation equivariance is already achieved in a regular CNN by weight sharing [92], rotational equivariance is not guaranteed. Recent studies show that rotational equivariance can actually be critical in data-driven SGS modeling [27,66,69,74]. To capture the rotational equivariance in the small-data regime, we propose two separate approaches: (1) DA, by including 3 additional rotated (by 90 • , 180 • , and 270 • ) counterparts of each original FDNS snapshot in the training set [74] and (2) by using a GCNN architecture, which enforces rotational equivariance by construction [92,93].…”

“…Past studies have shown that embedding physical insights or constraints can enhance the performance of data-driven models, e.g., in reduced-order models [e.g., [50][51][52][53][54][55][56][57] and in neural networks [e.g., 38,49,[58][59][60][61][62][63][64][65][66][67][68][69]. There are various ways to incorporate physics in neural networks (e.g., see the reviews by Kashinath et al [70], Balaji [71], and Karniadakis et al [72]).…”

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

“…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.…”

“…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. Their model was accurate and gave stable energy spectra for the various Reynolds numbers.…”