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This study proposes a novel approach for building surrogate models, in the form of reduced-order models(ROMs), for partial differential equation constrained optimization. A physics-constrained data-driven framework is developed to transform large-scale nonlinear optimization problems into ROM-constrained optimization problems. Unlike conventional methods, which suffer from instability of the forward sensitivity function, the proposed approach maps optimization problems to system dynamics optimization problems in Hilbert space to improve stability, reduce memory requirements, and lower computational cost. The utility of this approach is demonstrated for aerodynamic optimization of an NACA 0012 airfoil at $Re = 1000$ . A drag reduction of 9.35 % is obtained at an effective angle of attack of eight degrees, with negligible impact on lift. Similarly, a drag reduction of 20 % is obtained for fully separated flow at an angle of attack of $25^{\circ }$ . Results from these two optimization problems also reveal relationships between optimization in physical space and optimization of dynamical behaviours in Hilbert space.

This study proposes a new physics-constrained data-driven approach for sensitivity analysis and uncertainty quantification of large-scale chaotic Partial Differential Equations (PDEs). Unlike conventional sensitivity analysis, the proposed approach can manipulate the unsteady sensitivity function (i.e., tangent) for PDE-constrained optimizations. In this new approach, high-dimensional governing equations from physical space are transformed into an unphysical space (i.e., Hilbert space) to develop a closure model in the form of a Reduced-Order Model (ROM). This closure model is derived explicitly from the governing equations to set strong constraints on manifolds in Hilbert space. Afterward, a new data sampling method is proposed to build a data-driven approach for this framework. A series of least squares minimizations are set in the form of a novel auto-encoder system to solve this closure model. To compute sensitivities, least-squares shadowing minimization is applied to the ROM. It is shown that the proposed approach can capture sensitivities for large-scale chaotic dynamical systems, where finite difference approximations fail.

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