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

## TL;DR

This paper introduces a novel solver for nonlinear PDE-constrained optimization that combines SQP, POD, and neural networks to enable faster online computations for high-dimensional problems, demonstrated on the viscous Burgers' equation.

## Contribution

It presents a new integrated approach using neural networks with model reduction techniques for efficient PDE-constrained optimization.

## Key findings

- Significant online speed-up compared to traditional methods.
- Offline phase is longer but results in faster online evaluations.
- Reduced accuracy is a trade-off for increased computational speed.

## Abstract

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.