Sam Otto, AI Institute in Dynamic Systems, University of Washington

Model Reduction and Scientific Machine Learning for Continuum Mechanics

A reduced-order model (ROM) is a simplified approximation of a high-dimensional dynamical system which can be used for qualitative analysis, real-time forecasting, state estimation, and control. Shear-dominated fluid flows can be especially difficult to model using data-driven techniques such as 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. We evaluate this method against standard techniques on a nonlinear axisymmetric jet flow simulation with 100,000 state variables. Nonlinear extensions based on kernel methods and autoencoders are discussed, as well as prospects for techniques that do not require adjoints and are capable of being transferred to new spatial domains.

Sam Otto, AI Institute in Dynamic Systems, University of Washington

Model Reduction and Scientific Machine Learning for Continuum Mechanics

A reduced-order model (ROM) is a simplified approximation of a high-dimensional dynamical system which can be used for qualitative analysis, real-time forecasting, state estimation, and control. Shear-dominated fluid flows can be especially difficult to model using data-driven techniques such as 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. We evaluate this method against standard techniques on a nonlinear axisymmetric jet flow simulation with 100,000 state variables. Nonlinear extensions based on kernel methods and autoencoders are discussed, as well as prospects for techniques that do not require adjoints and are capable of being transferred to new spatial domains.