Modern decision-making in complex physical and engineered systems increasingly requires quantifying high-dimensional uncertainty and solving risk-averse optimization problems under uncertainty, often in near real time. Operator learning has emerged as a promising framework for constructing scalable surrogate models in this setting. Yet approximation errors in learned operators can significantly degrade the quality of downstream decisions arising from inverse problems and optimization tasks. In this talk, we present a framework that formulates operator learning through the lens of downstream optimization objectives. By deriving a priori error bounds for inverse problems and optimization under uncertainty, we motivate learning formulations that explicitly penalize errors in the derivatives of input–output maps. We illustrate the effectiveness of these ideas on applications including structural health monitoring, shape optimization, and fluid flow control. The resulting methods yield improved accuracy in surrogate-based optimization and consistently stronger empirical performance in related statistical learning tasks.