One-Shot Learning of Surrogates in PDE-Constrained Optimization under Uncertainty

IF 2.1 3区工程技术 Q2 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Siam-Asa Journal on Uncertainty Quantification Pub Date : 2024-06-12 DOI:10.1137/23m1553170

Philipp A. Guth, Claudia Schillings, Simon Weissmann

引用次数: 0

Abstract

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 2, Page 614-645, June 2024.
Abstract.We propose a general framework for machine learning based optimization under uncertainty. Our approach replaces the complex forward model by a surrogate, which is learned simultaneously in a one-shot sense when solving the optimal control problem. Our approach relies on a reformulation of the problem as a penalized empirical risk minimization problem for which we provide a consistency analysis in terms of large data and increasing penalty parameter. To solve the resulting problem, we suggest a stochastic gradient method with adaptive control of the penalty parameter and prove convergence under suitable assumptions on the surrogate model. Numerical experiments illustrate the results for linear and nonlinear surrogate models.

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不确定性条件下 PDE 受限优化中代用物的一次性学习

SIAM/ASA 不确定性量化期刊》，第 12 卷，第 2 期，第 614-645 页，2024 年 6 月。摘要：我们提出了一个基于机器学习的不确定性优化通用框架。我们的方法用代用模型取代了复杂的前向模型，在求解最优控制问题时，代用模型在一次学习的意义上被同时学习。我们的方法依赖于将问题重新表述为一个受惩罚的经验风险最小化问题，我们从大数据和增加惩罚参数的角度对该问题进行了一致性分析。为了解决由此产生的问题，我们提出了一种对惩罚参数进行自适应控制的随机梯度法，并证明了在代用模型的适当假设条件下的收敛性。数值实验说明了线性和非线性代用模型的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Siam-Asa Journal on Uncertainty Quantification Mathematics-Statistics and Probability

CiteScore

3.70

自引率

0.00%

发文量

期刊介绍： SIAM/ASA Journal on Uncertainty Quantification (JUQ) publishes research articles presenting significant mathematical, statistical, algorithmic, and application advances in uncertainty quantification, defined as the interface of complex modeling of processes and data, especially characterizations of the uncertainties inherent in the use of such models. The journal also focuses on related fields such as sensitivity analysis, model validation, model calibration, data assimilation, and code verification. The journal also solicits papers describing new ideas that could lead to significant progress in methodology for uncertainty quantification as well as review articles on particular aspects. The journal is dedicated to nurturing synergistic interactions between the mathematical, statistical, computational, and applications communities involved in uncertainty quantification and related areas. JUQ is jointly offered by SIAM and the American Statistical Association.