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Model Reduction for High-Dimensional Nonlinear Systems using State and Randomized Gradient Data

Model Reduction for High-Dimensional Nonlinear Systems using State and Randomized Gradient Data

We introduce data-driven techniques for constructing accurate, simplified reduced-order models (ROMs) of high-dimensional nonlinear dynamical systems, focusing primarily on fluid dynamics. These models are useful for tasks such as real-time forecasting, sensing, and control. Shear-dominated fluid flows can be especially difficult to model using data-driven techniques like proper orthogonal decomposition (a.k.a., principal component analysis), kernel-based manifold learning, and autoencoders because these methods discard low-variance variables, neglecting their importance for future dynamics. We show that this is a fundamental limitation related to the curse of dimensionality, and that additional information is needed to capture these sensitivity mechanisms. To extract reliable coordinates for forecasting, we introduce an efficient algorithm called Covariance Balancing Reduction using Adjoint Snapshots (CoBRAS). This method relies on state and randomized gradient data obtained by solving linearized adjoint equations to construct an oblique projection balancing the eﬀects of state variance and the sensitivity of future outputs to the truncated degrees of freedom. To refine an initial linear projection, such as that from CoBRAS, we introduce Trajectory-based Optimization for Oblique Projections (TrOOP) — a gradient descent method that minimizes forecasting error on trajectory data. We also develop a kernel-based extension of CoBRAS to extract powerful nonlinear coordinates. This method works implicitly with state and randomized gradient data appropriately lifted into reproducing kernel Hilbert spaces. We demonstrate these techniques and the limitations of standard methods on a nonlinear axisymmetric jet ﬂow simulation with 100,000 state variables.

This talk is based primarily on our paper:

S. E. Otto, A. Padovan, and C. W. Rowley, “Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance”, 2023, SIAM Journal on Scientific Computing, Vol. 45, No. 5

## Department of Mathematics and Statistics