Deep Uzawa for PDE constrained optimisation

2024/10/22 by Charalambos Makridakis, Makridakis, Charalambos G., Aaron Pim +3 · 3 citations
Computer Science · Engineering · #65K10 #Advanced Control Systems Optimization #Engineering Applied Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2410.17359

openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this work, we present a numerical solver for optimal control problems constrained by linear and semi-linear second-order elliptic PDEs. The approach is based on recasting the problem and includes an extension of Uzawa's algorithm to build approximating sequences for these constrained optimal control problems. We prove strong convergence of the iterative scheme in their respective norms, and this convergence is generalised to a class of restricted function spaces. We showcase the algorithm by demonstrating its use numerically with neural network methods that we coin Deep Uzawa Algorithms and show they perform favourably compared with some existing Deep Neural Network approaches.

Cited by

• Deep Uzawa for Kinetic Transport with Lagrange-Enforced Boundaries
• PINN-DG: Residual neural network methods trained with Finite Elements
• Neural Measures for learning distributions of Random PDEs

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