基于近似方向衍生物的 ReLU 神经网络 PDE 最佳控制后裔算法

IF 2.6 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Optimization Pub Date : 2024-07-02 DOI:10.1137/22m1534420

Guozhi Dong, Michael Hintermüller, Kostas Papafitsoros

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引用次数: 0

摘要

SIAM 优化期刊》，第 34 卷第 3 期，第 2314-2349 页，2024 年 9 月。摘要我们提出并分析了一种数值算法，用于求解一类学习信息半线性偏微分方程（PDE）的最优控制问题。这类偏微分方程包含原则上未知的成分，由非光滑 ReLU 神经网络近似。我们首先表明，以使用经典数值求解器为目的直接平滑 ReLU 网络会有一些缺点，例如可能会为相应的 PDE 引入多个解。这促使我们设计出一种数值算法，通过采用受无束法启发的下降算法，直接处理非平滑最优控制问题。我们提供了几个数值示例，并展示了该算法的效率。

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A Descent Algorithm for the Optimal Control of ReLU Neural Network Informed PDEs Based on Approximate Directional Derivatives

SIAM Journal on Optimization, Volume 34, Issue 3, Page 2314-2349, September 2024.
Abstract. We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs contain constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that direct smoothing of the ReLU network with the aim of using classical numerical solvers can have disadvantages, such as potentially introducing multiple solutions for the corresponding PDE. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown.

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来源期刊

SIAM Journal on Optimization 数学-应用数学

CiteScore

5.30

自引率

9.70%

发文量

101

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Optimization contains research articles on the theory and practice of optimization. The areas addressed include linear and quadratic programming, convex programming, nonlinear programming, complementarity problems, stochastic optimization, combinatorial optimization, integer programming, and convex, nonsmooth and variational analysis. Contributions may emphasize optimization theory, algorithms, software, computational practice, applications, or the links between these subjects.