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

建模仿真的一大前景是，模型可以作为复杂物理系统最佳决策的基础。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优化问题。在通常情况下，模型采用偏微分方程（PDEs）的形式，例如表征流体流动、固体力学、传热、声学和电磁学，这些问题被称为不确定性下的偏微分方程约束优化（PDE-OUU）。最近所谓的神经算子（NOs）的发展有望通过用快速评估的机器学习代理取代PDE模型来克服PDE- ouu问题的顽固性。该项目正在为NOs的构建和培训开发一个新的综合框架，以便它们更好地捕捉参数和决策空间的数学结构及其对模型输出的影响，从而推动不确定情况下的决策。这些NOs将在具有重大社会或技术重要性的不确定性问题下，在广泛的模型预测决策中实现PDE-OUU问题的可扩展、高效和准确的解决方案。这些问题的例子包括气候变化和自然灾害缓解、新材料的设计、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，在这个项目中开发的所有软件都将以开源的形式发布，以现有的成功的库（如hIPPYlib）为基础。两名博士生正在接受科学机器学习，随机优化和pde约束优化的跨学科接口的培训。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，或者当模型规模大且复杂时，PDE-OUU问题通常是难以解决的。然而，目前许多构建NOs的方法，以及训练它们的随机优化方法，都没有利用底层模型的数学特性，因此不够精确，无法作为OUU中偏微分方程的代理，特别是当训练数据由于获得它们的费用而受到限制时。为了从联合不确定参数和决策变量输入空间中利用pde控制的地图的数学特性来为优化目标提供模型输出，该项目寻求提取地图的几何、平滑和内在低维的知识，以协同设计(1)训练损失公式，(2)神经结构，(3)用于训练的随机优化算法。通过联合参数决策空间测量的精度，训练数据所需的PDE解数更少，结果NOs将表现出更高的精度。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.