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: 2245077
• 负责人: Jay Gopalakrishnan
• 金额: $ 29.99万
• 依托单位: Portland State University
• 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=2245077&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.

科学技术中普遍存在且通常至关重要的任务是综合来自控制物理定律和噪声观测数据（例如传感器系统提供的数据）的信息，以优化本项目中涉及的重要数量，包括地下流。该项目的总体目标是开发一个数学上严格的框架，用于从给定的信息源中“学习”底层的复杂模型。 ， 尤其深度学习在容错应用中并不自动意味着它在错误敏感的科学任务中取得了成功，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，这可以说是下一代模拟工具不可或缺的功能。这需要整合来自不同领域的概念工具，例如数值和函数分析、机器学习、统计、优化和信息几何，该项目为此目的聚集了一支多元化的团队，并作为副产品创造了独特的教育。学生和年轻研究人员的框架。根据应用，控制物理定律是根据各种类型的参数相关偏微分方程 (PDE) 来表述的，部分观察到的感兴趣状态位于（或接近）遍历该方程时获得的所有解中。学习或优化此类状态归结为涉及许多（参数）变量函数的不适定逆问题，为了应对这些障碍，该项目制定了一个“学习”框架，作为由以下组成的假设类的非线性回归问题。采用残差类型损失函数来避免大量高保真训练样本的昂贵计算，然后由所谓的变分正确残差风险来保证。优化的任何阶段的损失都与物理相关度量中产生的估计误差成正比。变分正确性是通过基础偏微分方程的稳定变分公式来实现的。它们通常基于当前不断发展的过程。 （不连续）Petrov-Galerkin 方法由于双重范数的固有出现，这需要在高维参数环境中有效评估所得损失函数的新策略。此外，专门调整的梯度流将作为开发鲁棒集成的重要组成部分。优化/适应/规范化策略。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。