A novel spatiotemporal decomposition and identification of sparse equations for human brain deformation

Low-dimensional coherent patterns underlie the behavior of many complex dynamical systems. We propose a novel dynamic mode decomposition (DMD) framework to discover reduced-order models of complex physics from spatiotemporal data. This algorithm combines time-delay embedding and reduced-order spectral projections to transform the initial dynamics into a latent space where DMD can efficiently resolve their evolution in space and time. Here, the efficiency and accuracy of this approach, which we call time-augmented, space-contracted DMD (TASC-DMD), is demonstrated in several benchmark tests, and is then utilized to compactly characterize human brain deformation. Sparse identification of nonlinear dynamics (SINDy) is then employed to discover a parsimonious model for brain deformation in TASC coordinates. This integrated algorithm (TASC-SINDy) is trained on 4D strain data from in vivo tagged magnetic resonance imaging (tagged MRI) in 36 human subjects from a cohort of 45 subjects. A unique and sparse set of governing equations describing the temporal dynamics of the brain was discovered using only three generic modes. The TASC-SINDy model achieved exceptional dimensionality reduction and accurately predicted dynamic strain fields for the nine test subjects not used for training. This data-driven approach can systematically unravel dynamics and improve predictions in many complex physical systems.