LC-SVD-DLinear: A low-cost physics-based hybrid machine learning model for data forecasting using sparse measurements

TL;DR

A novel methodology that integrates singular value decomposition (SVD) with a shallow linear neural network for forecasting high resolution fluid mechanics data and presents a variant of the method, LC-HOSVD-DLinear, which combines a low-cost version of the high-order singular value decomposition (LC-HOSVD) algorithm with the DLinear network, designed for high-order data.

Abstract

This article introduces a novel methodology that integrates singular value decomposition (SVD) with a shallow linear neural network for forecasting high resolution fluid mechanics data. The method, termed LC-SVD-DLinear, combines a low-cost variant of singular value decomposition (LC-SVD) with the DLinear architecture, which decomposes the input features-specifically, the temporal coefficients-into trend and seasonality components, enabling a shallow neural network to capture the non-linear dynamics of the temporal data. This methodology uses under-resolved data, which can either be input directly into the hybrid model or downsampled from high resolution using two distinct techniques provided by the methodology. Working with under-resolved cases helps reduce the overall computational cost. Additionally, we present a variant of the method, LC-HOSVD-DLinear, which combines a low-cost version of the high-order singular value decomposition (LC-HOSVD) algorithm with the DLinear network, designed for high-order data. These approaches have been validated using two datasets: first, a numerical simulation of three-dimensional flow past a circular cylinder at $Re = 220$; and second, an experimental dataset of turbulent flow passing a circular cylinder at $Re = 2600$. The combination of these datasets demonstrates the robustness of the method. The forecasting and reconstruction results are evaluated through various error metrics, including uncertainty quantification. The work developed in this article will be included in the next release of ModelFLOWs-app

Figures (34)

• Figure 1: An example of preprocessed data, split into train, validation and test sets, scaled between -1 and 1 with min-max scaling.
• Figure 2: Illustration of a sliding window applied to the temporal coefficients $\mathbf{T}$, with dimensions $K \times N$. This process generates $W$ sequences $(\mathbf{T}_z)$, each of size $L \times N$, along with a horizon $(T_h)$ of size $H \times N$, with$H = 1$). Here, $w$ the sequence number, $L$ is the sequence length, $H$ denotes the horizon length, and $N$ is the number of SVD modes.
• Figure 3: Representation of the DLinear architecture, showing how an input sequence $T_z$ is decomposed into trend $T_t$ and seasonality $T_s$ with the use of an average pooling layer, and both components are passed to individual linear layers which predict the next trend $H_t$ and seasonality $H_s$ values, which are summed, resulting in the prediction $\hat{T}_h$.
• Figure 4: Outline of the LC-SVD-DLinear model, showing how the different parts of the hybrid model are interconnected to forecast high-resolution snapshots, given a low-resolution dataset.
• Figure 5: Streamwise (left), normal (middle) and spanwise (right) velocities of a snapshot of the three-dimensional Re $= 220$ cylinder dataset from Ref. VegaLeClaincheBook20 in the $XY$ plane, for $z = 32$.