Co-Design of Neural Operators and Stochastic Optimization Algorithms for Learning Surrogates for PDE-Constrained Optimization Under Uncertainty

建模和仿真的最大希望之一是，这些模型可以作为复杂物理系统的最优决策的基础。在许多情况下，这些系统的模型不是完全已知的，因此包含不确定的参数。这就产生了不确定条件下的最优化问题。在描述流体流动、固体力学、传热学、声学、电磁学等数学模型的偏微分方程组(PDE)形式的情况下，这些问题称为不确定条件下的偏微分方程组约束优化(PDE-OUU)。最近发展的所谓神经算子(Nos)有望通过用一种快速评估机器学习的代理来取代PDE模型来克服PDE-OUU问题的复杂性。该项目正在开发一个新的综合框架，用于构建和培训国家网络，以便它们更好地捕捉参数和决策空间的数学结构及其对在不确定情况下推动决策的模型产出的影响。这些NOS将使PDE-OUU问题能够在具有重大社会或技术重要性的不确定性问题下，通过广泛的模型预测决策实现可扩展、高效和准确的解决方案。这类问题的例子包括气候变化和自然灾害缓解、新材料设计、关键基础设施的运行、针对患者的疾病治疗规划和环境观测系统设计。为了促进这些算法的采用，本项目中开发的所有软件都将以开源形式发布，建立在现有的成功库(如hIPPYlib)的基础上。两名博士生正在接受科学机器学习、随机优化和PDE约束优化的跨学科界面培训。尽管PDE-OUU问题在许多技术、科学、工程和医学领域具有重要意义，但当不确定参数或决策变量维度较大时，或者当模型规模较大且复杂时，PDE-OUU问题通常是难以解决的。然而，当前用于构建NOS的许多方法以及用于训练它们的随机优化方法没有利用基础模型的数学属性，因此不足以用作OUU中的PDE的代理，特别是当训练数据由于获得它们的费用而受到限制时。为了利用从联合不确定参数和决策变量输入空间到模型输出的偏微分方程控制的映射的数学特性，该项目试图提取映射的几何、光滑性和内在低维的知识，以协同设计(1)训练损失公式，(2)神经结构和(3)用于训练的随机优化算法。由此产生的NOS将表现出更高的准确性，训练数据所需的PDE解更少，准确性通过联合参数决策空间测量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

One of the great promises of modeling & simulation is that the models can serve as a basis for optimal decision making for complex physical systems. In many cases, the models for these systems are not fully known, and as a result contain uncertain parameters. This gives rise to problems in optimization under uncertainty (OUU). In the common situation in which the models take the form of partial differential equations (PDEs), for example characterizing fluid flow, solid mechanics, heat transfer, acoustics, and electromagnetics, the problems are known as PDE-constrained optimization under uncertainty (PDE-OUU). The recent development of so-called neural operators (NOs) promises to overcome the intractability of PDE-OUU problems by replacing the PDE model with a rapid-to-evaluate machine-learned surrogate. This project is developing a new integrated framework for both construction and training of NOs so that they better capture the mathematical structure of parameter and decision space and their impact on model outputs that drive decision making under uncertainty. These NOs will enable scalable, efficient, and accurate solution of PDE-OUU problems across a broad range of model-predictive decision-making under uncertainty problems of great societal or technological importance. Examples of such problems include those in climate change and natural hazard mitigation, design of new materials, operation of critical infrastructure, patient-specific disease treatment planning, and environmental observing system design. To facilitate the adoption of these algorithms, all software developed in this project will be released in open source form, building on existing successful libraries such as hIPPYlib. Two PhD students are being trained at the interdisciplinary interfaces of scientific machine learning, stochastic optimization, and PDE-constrained optimization. Despite their great importance in many technological, scientific, engineering, and medical fields, PDE-OUU problems are typically intractable when the uncertain parameter or decision variable dimensions are large, or when the models are large-scale and complex. However, many current methods for constructing NOs, as well as stochastic optimization methods to train them, do not exploit mathematical properties of the underlying models and as such are not sufficiently accurate to serve as proxies for the PDEs in OUU, particularly when the training data are limited due to the expense of obtaining them. To exploit mathematical properties of the PDE-governed maps from joint uncertain parameter and decision variable input space to model outputs that inform the optimization objective, this project seeks to extract knowledge of the geometry, smoothness, and intrinsic low dimensionality of the maps to synergistically co-design (1) training loss formulations, (2) neural architectures, and (3) stochastic optimization algorithms for training. The resulting NOs will exhibit greater accuracy with fewer PDE solves needed for training data, with accuracy measured over joint parameter–decision space.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.