FRG: Collaborative Research: Variationally Stable Neural Networks for Simulation, Learning, and Experimental Design of Complex Physical Systems

FRG：协作研究：用于复杂物理系统仿真、学习和实验设计的变稳定神经网络

• 批准号: 2245147
• 负责人: Leszek Demkowicz
• 金额: $ 30万
• 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=2245147&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Variationally; Stable

A ubiquitous and often critical task in science and technology is to synthesize information from governing physical laws and noisy observational data, such as those provided by sensor systems, in order to optimize important quantities of interest. Examples touched upon in this project include subsurface flow through porous media, fiber optics, waveguide design, and material science applications. The overarching goal of this project is to develop a mathematically rigorous framework for “learning” the underlying complex models from the given sources of information. The recent stunning successes of modern machine learning, especially deep learning, in error-tolerant applications, does not automatically imply its success in error-sensitive scientific tasks. Targeting the latter, this project aims to significantly advance prediction capabilities through rigorous accuracy quantification and certification, arguably an indispensable feature of next generation simulation tools in science and technology. This requires integrating conceptual tools from diverse areas such as numerical and functional analysis, machine learning, statistics, optimization, and information geometry. The project gathers a diverse team for this purpose, and as a byproduct, creates a unique educational framework for students and young researchers. Governing physical laws are formulated in terms of (systems of) parameter dependent partial differential equations (PDEs) of various types depending on the application. Partially observed states of interest are then among (or close to) all those solutions that are obtained when traversing the parameter space. Learning or optimizing such states boils down to ill-posed inverse problems involving functions of many (parametric) variables. To cope with these obstructions, this project formulates a “learning” framework as a nonlinear regression problem over hypothesis classes comprised of deep neural networks. Residual type loss functions are employed to avoid expensive computation of a large number of high-fidelity training samples. Accuracy quantification and a posteriori certification is then warranted by so-called variationally-correct residual risks. This means that the size of the loss at any stage of the optimization is uniformly proportional to the error incurred by the resulting estimation in a physically relevant metric. The variational correctness is achieved through stable variational formulations of the underlying PDEs. They are typically based on currently evolving (discontinuous) Petrov-Galerkin methodologies. Due to the inherent appearance of dual norms, this requires new strategies for efficiently evaluating resulting loss functions in the high dimensional parametric context. Moreover, specially adapted gradient flows will serve as an important constituent in developing robust integrated optimization/adaptation/regularization strategies.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.

科学和技术中无处不在且往往至关重要的任务是从管理物理定律和噪声观察数据（例如传感器系统提供的数据）中综合信息，以优化重要的关注数量。该项目中涉及的示例包括通过多孔介质，光纤，波导设计和材料科学应用的地下流。该项目的总体目标是开发一个数学上严格的框架，以从给定的信息来源“学习”基础复杂模型。最近的现代机器学习的惊人成功，尤其是深度学习，在易于错误的应用中，并不意味着它在错误敏感的科学任务中取得了成功。针对以后的目标，该项目旨在通过严格的准确性量化和认证来显着提高预测能力，这可以说是科学和技术中下一代模拟工具的必不可少的功能。这需要集成来自潜水区域的概念工具，例如数值和功能分析，机器学习，统计，优化和信息几何形状。该项目为此目的聚集了潜水员团队，作为副产品，为学生和年轻的研究人员创建了独特的教育框架。管理物理定律是根据依赖于应用程序的各种类型的参数依赖性偏微分方程（PDE）来制定的。然后，部分观察到的感兴趣状态在（或接近）穿越参数空间时获得的所有解决方案。学习或优化此类状态归结为涉及许多（参数）变量功能的不良反问题。为了应对这些对象，该项目将“学习”框架制定为包括深神经网络的假设类别的非线性回归问题。使用残留类型损失功能，以避免对大量高保真训练样本进行昂贵的计算。然后，应由所谓的变异余量剩余风险来保证准确性定量和后验认证。这意味着优化的任何阶段的损失的大小与在物理相关度量的估计中所产生的误差均匀成正比。通过基础PDE的稳定变异公式实现变异正确性。它们通常基于当前不断发展的（不连续的）Petrov-Galerkin方法。由于双重规范的继承外观，这需要有效评估高维参数上下文中产生的损失函数的新策略。此外，特别适应的梯度流将成为开发可靠的综合优化/适应/正则化策略的重要组成部分。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子优点和更广泛的审查标准评估来诚实地获得支持。