# Toward real-time optimization through model reduction and model discrepancy sensitivities

## TL;DR

This paper introduces a method combining hyper-differential sensitivity analysis with model discrepancy to enhance the reliability of reduced-order model constrained optimization, enabling near real-time solutions for complex systems.

## Contribution

It proposes a novel HDSA-MD approach that improves ROM-based optimization accuracy with minimal full-order model evaluations during online phases.

## Key findings

- HDSA-MD significantly improves optimization solutions.
- Only one full-order model evaluation needed online.
- Effective in atmospheric contaminant control and wildfire estimation.

## Abstract

Optimization problems arise in a range of scenarios, from optimal control to model parameter estimation. In many applications, such as the development of digital twins, it is essential to solve these optimization problems within wall-clock-time limitations. However, this is often unattainable for complex systems, such as those modeled by nonlinear partial differential equations. One strategy for mitigating this issue is to construct a reduced-order model (ROM) that enables more rapid optimization. In particular, the use of nonintrusive ROMs -- those that do not require access to the full-order model at evaluation time -- is popular because they facilitate optimization solutions can be computed within the wall-clock-time requirements. However, the optimization solution will be unreliable if the iterates move outside the ROM training data. This article proposes the use of hyper-differential sensitivity analysis with respect to model discrepancy (HDSA-MD) as a computationally efficient tool to augment ROM-constrained optimization and improve its reliability. The proposed approach consists of two phases: (i) an offline phase where several full-order model evaluations are computed to train the ROM, and (ii) an online phase where a ROM-constrained optimization problem is solved, $N=\mathcal{O}(1)$ full-order model evaluations are computed, and HDSA-MD is used to enhance the optimization solution using the full-order model data. Numerical results are demonstrated for two examples, atmospheric contaminant control and wildfire ignition location estimation, in which a ROM is trained offline using inaccurate atmospheric data. The HDSA-MD update yields a significant improvement in the ROM-constrained optimization solution using only one full-order model evaluation online with corrected atmospheric data.