# Reduced Order Modeling for Nonlinear PDE-constrained Optimization using Neural Networks

arXiv: https://arxiv.org/abs/1904.06965 · HTML: https://mlsanity.com/paper/1904.06965/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2019.

## Built on by (most cited first, 2 of 2)

- [Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solver](https://mlsanity.com/paper/2105.11045/) (2021)
- [A two stages Deep Learning Architecture for Model Reduction of Parametric Time-Dependent Problems](https://mlsanity.com/paper/2301.09926/) (2023)

## Abstract (arXiv)

Nonlinear model predictive control (NMPC) often requires real-time solution to optimization problems. However, in cases where the mathematical model is of high dimension in the solution space, e.g. for solution of partial differential equations (PDEs), black-box optimizers are rarely sufficient to get the required online computational speed. In such cases one must resort to customized solvers. This paper present a new solver for nonlinear time-dependent PDE-constrained optimization problems. It is composed of a sequential quadratic programming (SQP) scheme to solve the PDE-constrained problem in an offline phase, a proper orthogonal decomposition (POD) approach to identify a lower dimensional solution space, and a neural network (NN) for fast online evaluations. The proposed method is showcased on a regularized least-square optimal control problem for the viscous Burgers' equation. It is concluded that significant online speed-up is achieved, compared to conventional methods using SQP and finite elements, at a cost of a prolonged offline phase and reduced accuracy.