Decomposed sparse modal optimization: Interpretable reduced-order modeling of unsteady flows

Modal analysis plays a crucial role in fluid dynamics, offering a powerful tool for reducing the complexity of high-dimensional fluid flow data while extracting meaningful insights into flow physics. This is particularly important in the study of cardiovascular flows, where modal techniques help characterize unsteady flow structures, improve reduced-order modeling, and inform disease diagnosis and rapid medical device design. The most commonly used method, proper orthogonal decomposition (POD), is highly interpretable but suffers from its linearity, which limits its ability to capture nonlinear interactions. In this work, we introduce decomposed sparse modal optimization (DESMO), a nonlinear, adaptive extension of POD that improves the accuracy of flow field reconstruction while requiring fewer modes. We use modern gradient descent-based optimization tools to optimize the spatial modes and temporal coefficients concurrently while using a sparsity-promoting loss term. We demonstrate these on a canonical fluid flow benchmark, flow over a cylinder, a real-world example, blood flow inside a brain aneurysm, and a turbulent channel flow. DESMO can identify spatial modes that resemble higher-order POD modes while uncovering entirely new spatial structures in some cases. Different versions of DESMO can leverage Fourier series for modeling temporal coefficients, an autoencoder for spatial mode optimization, and symbolic regression for discovering differential equations for temporal evolution. Our results demonstrate that DESMO not only provides a more accurate representation of fluid flows but also preserves the interpretability of classical POD by having an analytical modal decomposition equation, offering a promising approach for reduced-order modeling across engineering applications.