Data-Driven Modeling of Turbulent Flows: Stochastic Koopman Models and Nonlinear Modal Energetics

Predictive, reduced-order models for turbulent flows must confront two core difficulties: the dynamics are strongly nonlinear and multi-scale, and the resulting trajectories are only predictable over limited horizons, even when the governing equations are known. In this talk I will present two complementary data-driven advances that address these issues from different angles: (i) stable, uncertainty-aware forecasting with a stochastic Koopman reduced-order model, and (ii) an interpretable decomposition that reveals where, how, and between which structures nonlinear energy transfer occurs. On the modeling side, I will outline the stochastic low-dimensional inflated convolutional Koopman (SLICK) model, which evolves SPOD-based convolutional (time-delay) coordinates with a finite-dimensional Koopman operator and closes unresolved nonlinear interactions by augmenting the state and modeling the residual as a stochastic source. To complement forecasting with mechanism, I will then introduce Triadic Orthogonal Decomposition (TOD), which identifies coherent structures that optimally capture spectral momentum transfer, ranks and quantifies triadic coupling via an energy-budget bispectrum, and distinguishes recipient, donor, and catalyst roles while localizing interaction regions (shown for the cylinder wake, wind-turbine wake data, and isotropic turbulence). Together, these two results illustrate a theme for realistic turbulent flows: data-driven reduced-order modeling that is both predictive and physically interpretable through nonlinear energy-transfer structure.