# Progressive construction of a parametric reduced-order model for PDE-constrained optimization

arXiv: https://arxiv.org/abs/1407.7618 · HTML: https://mlsanity.com/paper/1407.7618/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2014.

## Built on by (most cited first, 13 of 13)

- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations](https://mlsanity.com/paper/2402.14646/) (2024)
- [Reduced-order modeling for parameterized PDEs via implicit neural representations](https://mlsanity.com/paper/2311.16410/) (2023)
- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [Learning To Solve Differential Equation Constrained Optimization Problems](https://mlsanity.com/paper/2410.01786/) (2024)
- [A surrogate model for topology optimisation of elastic structures via parametric autoencoders](https://mlsanity.com/paper/2507.22539/) (2025)
- [Speeding up Computational Morphogenesis with Online Neural Synthetic Gradients](https://mlsanity.com/paper/2104.12282/) (2021)
- [Neural-POD: A Plug-and-Play Neural Operator Framework for Infinite-Dimensional Functional Nonlinear Proper Orthogonal Decomposition](https://mlsanity.com/paper/2602.15632/) (2026)
- [Learning to Solve Optimization Problems Constrained with Partial Differential Equations](https://mlsanity.com/paper/2509.24573/) (2025)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [Parametric Nonlinear Volterra Series via Machine Learning: Transonic Aerodynamics](https://mlsanity.com/paper/2410.19514/) (2024)
- [Active learning for data-driven reduced models of parametric differential systems with Bayesian operator inference](https://mlsanity.com/paper/2601.00038/) (2025)

## Abstract (arXiv)

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%.