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The paper unrolls an inexact Uzawa method into a trainable network, replacing classical preconditioners and PDE solves with learned components, and claims universal approximation and asymptotic ε-optimality for nonsmooth elliptic and parabolic control problems. For ML readers, the useful question is whether the speedups hold outside the authors' benchmarks and how the learned components generalize across problem instances. The hybrid model-plus-learning design is a practical pattern for PDE-constrained optimization, but the "real-time" claim needs checking against realistic problem sizes.
Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEs
arXiv:2602.12273v2 Announce Type: replace-cross Abstract: We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic $\varepsilon$-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.
Abstract:We propose an optimization-informed deep neural network approach, named iUzawa-Net, aiming for the first solver that enables real-time solutions for a class of nonsmooth optimal control problems of linear partial differential equations (PDEs). The iUzawa-Net unrolls an inexact Uzawa method for saddle point problems, replacing classical preconditioners and PDE solvers with specifically designed learnable neural networks. We prove universal approximation properties and establish the asymptotic $\varepsilon$-optimality for the iUzawa-Net, and validate its promising numerical efficiency through nonsmooth elliptic and parabolic optimal control problems. Our techniques offer a versatile framework for designing and analyzing various optimization-informed deep learning approaches to optimal control and other PDE-constrained optimization problems. The proposed learning-to-control approach synergizes model-based optimization algorithms and data-driven deep learning techniques, inheriting the merits of both methodologies.
Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2602.12273 [math.OC]
(or arXiv:2602.12273v2 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2602.12273 Focus to learn more arXiv-issued DOI via DataCite
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