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

不确定性下偏微分方程约束优化学习代理的神经算子和随机优化算法的协同设计

• 批准号: 2324643
• 负责人: Vijaya Raghaven Bollapragada
• 金额: $ 49.98万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2324643&HistoricalAwards=false
• 关键词: Co; Design; Neural; Operators; Stochastic

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.

建模和仿真的巨大承诺之一是，这些模型可以作为对复杂物理系统进行最佳决策的基础。在许多情况下，这些系统的模型尚不完全了解，因此包含不确定的参数。这引起了不确定性（OUU）优化问题的问题。在模型采用偏微分方程（PDE）形式的常见情况下，例如表征流体流动，固体力学，传热，声学和电子设备，这些问题被称为不确定性下的PDE受限优化（PDE-OUU）。所谓的神经操作员（NOS）的最新发展有望通过用快速评估的机器学习的替代替换PDE模型来克服PDE-OUU问题的难关性。该项目正在开发一个新的集成框架，用于构建和培训NOS，以便它们更好地捕获参数和决策空间的数学结构及其对模型输出的影响，从而使决策在不确定性下推动决策。这些NOS将在不确定性问题的不确定性问题下，在广泛的模型预测决策中启用可扩展，高效和准确的PDE-OUU问题解决方案。此类问题的例子包括在气候变化和减轻自然危害的情况下，新材料的设计，关键基础设施的运作，特定于患者的疾病治疗计划以及环境观察系统设计。为了促进这些算法的采用，本项目中开发的所有软件将以开源形式发布，以Hippylib等现有成功的库为基础。在科学机器学习，随机优化和PDE受限优化的跨学科界面上，正在对两名博士学位学生进行培训。尽管在许多技术，科学，工程和医学领域中，当不确定的参数或决策变量尺寸较大时，或者模型大规模且复杂时，PDE-OUU问题通常是可悲的。 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, To explore 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地图与协同共同设计（1）训练损失公式，（2）神经体系结构以及（3）训练的随机优化算法的几何形状，平滑度和内在的低维（1）训练损失公式。由培训数据所需的PDE解决方案较少，最终的PDE求解将执行更高的准确性，并且根据联合参数 - 任务空间的精确度来衡量。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的评估来评估的支持标准。