Sub-grid modelling for two-dimensional turbulence using neural networks

In this investigation, a data-driven turbulence closure framework is introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the sub-grid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a-priori and a-posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability density function based validation of sub-grid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for sub-grid quantity inference. In addition, it is also observed that some measure of a-posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data

A quantum approach for digital signal processing

We propose a novel quantum approach to signal processing, including a quantum algorithm for low-pass and high-pass filtering, based on the sequency-ordered Walsh-Hadamard transform. We present quantum circuits for performing the sequency-ordered Walsh-Hadamard transform, as well as quantum circuits for low-pass, high-pass, and band-pass filtering. Additionally, we provide a proof of correctness for the quantum circuit designed to perform the sequency-ordered Walsh-Hadamard transform. The performance and accuracy of the proposed approach for signal filtering were illustrated using computational examples, along with corresponding quantum circuits, for DC, low-pass, high-pass, and band-pass filtering. Our proposed algorithm for signal filtering has a reduced gate complexity and circuit depth of $O (\log_2 N)$, compared to at least $O ((\log_2 N )^2)$ associated with Quantum Fourier Transform (QFT) based filtering (excluding state preparation and measurement costs). In contrast, classical Fast Fourier Transform (FFT) based filtering approaches have a complexity of $O (N \log_2 N )$. This shows that our proposed approach offers a significant improvement over QFT-based filtering methods and classical FFT-based filtering methods. Such enhanced efficiency of our proposed approach holds substantial promise across several signal processing applications by ensuring faster computations and efficient use of resources via reduced circuit depth and lower gate complexity.Comment: 29 page

A hybrid classical-quantum algorithm for solution of nonlinear ordinary differential equations

A hybrid classical-quantum approach for the solution of nonlinear ordinary differential equations using Walsh-Hadamard basis functions is proposed. Central to this hybrid approach is the computation of the Walsh-Hadamard transform of arbitrary vectors, which is enabled in our framework using quantum Hadamard gates along with state preparation, shifting, scaling, and measurement operations. It is estimated that the proposed hybrid classical-quantum approach for the Walsh-Hadamard transform of an input vector of size N results in a considerably lower computational complexity (O(N) operations) compared to the Fast Walsh-Hadamard transform (O(N log2(N)) operations). This benefit will also be relevant in the context of the proposed hybrid classical-quantum approach for the solution of nonlinear differential equations. Comparisons of results corresponding to the proposed hybrid classical-quantum approach and a purely classical approach for the solution of nonlinear differential equations (for cases involving one and two dependent variables) were found to be satisfactory. Some new perspectives relevant to the natural ordering of Walsh functions (in the context of both classical and hybrid approaches for the solution of nonlinear differential equations) and representation theory of finite groups are also presented here.Comment: 29 pages, 10 figure

Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance

A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence

In the present study, we investigate different data-driven parameterizations for large eddy simulation of two-dimensional turbulence in the \emph{a priori} settings. These models utilize resolved flow field variables on the coarser grid to estimate the subgrid-scale stresses. We use data-driven closure models based on localized learning that employs multilayer feedforward artificial neural network (ANN) with point-to-point mapping and neighboring stencil data mapping, and convolutional neural network (CNN) fed by data snapshots of the whole domain. The performance of these data-driven closure models is measured through a probability density function and is compared with the dynamic Smagorinksy model (DSM). The quantitative performance is evaluated using the cross-correlation coefficient between the true and predicted stresses. We analyze different frameworks in terms of the amount of training data, selection of input and output features, their characteristics in modeling with accuracy, and training and deployment computational time. We also demonstrate computational gain that can be achieved using the intelligent eddy viscosity model that learns eddy viscosity computed by the DSM instead of subgrid-scale stresses. We detail the hyperparameters optimization of these models using the grid search algorithm

Data-driven deconvolution for large eddy simulations of Kraichnan turbulence

In this article, we demonstrate the use of artificial neural networks as optimal maps which are utilized for convolution and deconvolution of coarse-grained fields to account for sub-grid scale turbulence effects. We demonstrate that an effective eddy-viscosity is predicted by our purely data-driven large eddy simulation framework without explicit utilization of phenomenological arguments. In addition, our data-driven framework precludes the knowledge of true sub-grid stress information during the training phase due to its focus on estimating an effective filter and its inverse so that grid-resolved variables may be related to direct numerical simulation data statistically. The proposed predictive framework is also combined with a statistical truncation mechanism for ensuring numerical realizability in an explicit formulation. Through this we seek to unite structural and functional modeling strategies for modeling non-linear partial differential equations using reduced degrees of freedom. Both a priori and a posteriori results are shown for a two-dimensional decaying turbulence case in addition to a detailed description of validation and testing. A hyperparameter sensitivity study also shows that the proposed dual network framework simplifies learning complexity and is viable with exceedingly simple network architectures. Our findings indicate that the proposed framework approximates a robust and stable sub-grid closure which compares favorably to the Smagorinsky and Leith hypotheses for capturing the theoretical $k^{-3}$ scaling in Kraichnan turbulence