A physics-based deep learning framework for super-resolution in fluid dynamics based on higher-order singular value decomposition

Abstract

In this work, a novel methodology combining higher-order singular value decomposition (HOSVD) with deep learning is presented for the super-resolution (SR) of computational fluid dynamics and sparse experimental datasets. This approach, referred to as HOSVD-SR (The software is available at https://modelflows.github.io/modelflowsapp/), integrates modal decomposition with machine-learning techniques to create a physics-based hybrid model that enhances spatial resolution while preserving the dominant flow structures. The proposed methodology leverages HOSVD, a higher-order extension of SVD that is well suited to high-dimensional data arranged in the tensor form. HOSVD independently extracts the mode matrices and singular values associated with each tensor dimension, including the spatial and temporal dimensions. This dimension-wise treatment is physically motivated: by decomposing the data independently along each spatial direction, HOSVD identifies the energetically dominant and spatially organized flow structures while filtering low-energy noise and spatial redundancies. This is particularly relevant to the accurate reconstruction of laminar, transitional, and turbulent flows. Each low-resolution spatial mode matrix, together with its corresponding singular values, is provided as input to an independent decoder/autoencoder-type neural network trained to increase the spatial dimensionality of that mode matrix. The enhanced spatial mode matrices are subsequently combined with the unchanged component and temporal mode matrices and the core tensor to reconstruct the high-resolution flow database. The method, therefore, enhances resolution from spatial patterns extracted from a sequence of physically consistent flow fields rather than treating individual snapshots as independent images. The HOSVD-SR methodology is evaluated using three databases of different complexity: two- and three-dimensional numerical simulations of flow past a circular cylinder and a numerical database of flow in two axisymmetric concentric jets. For the cylinder-flow cases, HOSVD-SR accurately reconstructs the wake topology, including the von Kármán vortex street, near-body structures, and, where applicable, asymmetric and three-dimensional flow features. For the concentric-jets case, the method recovers the oscillating structures developing along the jet shear layers, even after severe spatial downsampling. Across the investigated datasets and compression ratios (CRs), HOSVD-SR produces lower reconstruction errors than the previously proposed SVD-SR methodology. For the numerical test cases in which deterministic interpolation is also evaluated, HOSVD-SR substantially outperforms the interpolation baseline, particularly at high CRs for which important flow structures are absent from the retained spatial samples. These results demonstrate the potential of the proposed physics-based framework as a new approach to the SR of fluid-dynamics data. Although additional developments concerning network optimization, computational efficiency, scalability, and validation for other flow configurations remain for future studies, the dimension-wise formulation makes the method potentially transferable to databases generated using other numerical and experimental approaches in fluid mechanics.