We propose a novel machine learning framework for solving optimization problems governed by large-scale partial differential equations (PDEs) with high-dimensional random parameters. Such optimization under uncertainty (OUU) problems may be computational prohibitive using classical methods, particularly when a large number of samples is needed to evaluate risk measures at every iteration of an optimization algorithm, where each sample requires the solution of an expensive-to-solve PDE. To address this challenge, we propose a new neural operator approximation of the PDE solution operator that has the combined merits of (1) accurate approximation of not only the map from the joint inputs of random parameters and optimization variables to the PDE state, but also its derivative with respect to the optimization variables, (2) efficient construction of the neural network using reduced basis architectures that are scalable to high-dimensional OUU problems, and (3) requiring only a limited number of training data to achieve high accuracy for both the PDE solution and the OUU solution. We refer to such neural operators as multi-input reduced basis derivative informed neural operators (MR-DINOs). We demonstrate the accuracy and efficiency our approach through several numerical experiments, i.e. the risk-averse control of a semilinear elliptic PDE and the steady state Navier--Stokes equations in two and three spatial dimensions, each involving random field inputs. Across the examples, MR-DINOs offer $10^{3}$--$10^{7} \times$ reductions in execution time, and are able to produce OUU solutions of comparable accuracies to those from standard PDE based solutions while being over $10 \times$ more cost-efficient after factoring in the cost of construction.

翻译：我们提出了一种新颖的机器学习框架，用于求解受大规模偏微分方程（PDEs）支配且具有高维随机参数的优化问题。这类不确定性下的优化（OUU）问题若采用经典方法可能计算成本极高，特别是当优化算法每次迭代都需要大量样本来评估风险度量时，且每个样本均需求解计算昂贵的PDE。为应对这一挑战，我们提出了一种新的神经算子来逼近PDE解算子，其兼具以下优势：（1）不仅能精确逼近从随机参数与优化变量的联合输入到PDE状态的映射，还能精确逼近该映射关于优化变量的导数；（2）采用可扩展至高维OUU问题的降阶基架构高效构建神经网络；（3）仅需有限训练数据即可同时实现PDE解与OUU解的高精度逼近。我们将此类神经算子称为多输入降阶基导数信息神经算子（MR-DINOs）。通过多个数值实验验证了所提方法的准确性与效率，包括风险规避控制下的半线性椭圆型PDE以及二维和三维空间中的稳态Navier-Stokes方程，每个问题均涉及随机场输入。在所有算例中，MR-DINOs将执行时间降低了$10^{3}$–$10^{7}$倍，且能在考虑构建成本后，以超过标准PDE求解方法10倍以上的成本效益，获得精度相当的OUU解。