We propose a general framework for machine learning based optimization under uncertainty. Our approach replaces the complex forward model by a surrogate, e.g., a neural network, 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.

翻译：我们提出了一种基于机器学习的通用不确定性优化框架。该方法用代理模型（如神经网络）替代复杂正演模型，在求解最优控制问题时以一步学习方式同步进行训练。通过将问题重新表述为带惩罚项的经验风险最小化问题，我们为大数据和递增惩罚参数下的模型一致性提供了理论分析。为解决该问题，我们提出了带惩罚参数自适应控制的随机梯度方法，并在代理模型的合理假设下证明了收敛性。数值实验展示了线性和非线性代理模型的结果。