Neural Networks for Stationary and Evolutionary Variational Problems

用于稳态和进化变分问题的神经网络

• 批准号: 2307273
• 负责人: Stephan Wojtowytsch
• 金额: $ 20.96万
• 依托单位: Texas A&M 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: 2023-08-01 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307273&HistoricalAwards=false
• 关键词: Neural; Networks; Stationary; Evolutionary; Variational

Artificial neural networks have become one of the dominant models in data science, used in applications from image classification to natural language processing. Their empirical success in these diverse fields has sparked interest in further applying such models in new directions, such as numerical analysis and scientific computing. This project is aimed at developing a deeper understanding of the capabilities and limitations of the role of neural network models used for numerical analysis and scientific computing, particularly when compared with more classic tools. This is essential in enabling neural network models to be widely deployed in sensitive fields across engineering domains. Graduate students will be trained as part of this project, modern tools from data science and deep learning will be incorporated into graduate curricula, and outreach activities are planned to attract undergraduates as well as underrepresented groups in STEM into this research area.The focus of this work is on the use of neural network models in numerical algorithms used in models based on the calculus of variations, targeting two case studies. The first is related to functionals that exhibit the Lavrentiev gap phenomena, where an energy gap between the lowest energy achievable by shallow neural networks and more general functions is considered. The second is the Allen-Cahn equation, where the solution strategy of physics-inspired neural networks is analyzed. In the second problem, the adaptivity of neural networks to low-dimensional moving interfaces plays a key role when comparing to e.g. fixed mesh finite element methods. The theoretical results are intended to better understand two fundamental challenges. The first is whether the adaptivity of neural networks can be harnessed for the numerical approximation of spatially very inhomogeneous variational problems. The second seeks to understand the precise situations in which neural network solvers are not expected to outperform traditional solvers.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.

人工神经网络已经成为数据科学的主要模型之一，应用于从图像分类到自然语言处理的各个领域。他们在这些不同领域的经验成功激发了对进一步将这些模型应用于新方向的兴趣，例如数值分析和科学计算。该项目旨在深入了解用于数值分析和科学计算的神经网络模型的功能和局限性，特别是与更经典的工具相比。这对于使神经网络模型在工程领域的敏感领域得到广泛应用至关重要。作为该项目的一部分，研究生将接受培训，数据科学和深度学习的现代工具将纳入研究生课程，并计划开展外展活动，以吸引本科生以及STEM中代表性不足的群体进入这一研究领域。这项工作的重点是在基于变分法的模型中使用的数值算法中使用神经网络模型，针对两个案例研究。第一个与表现出Lavrentiev间隙现象的函数有关，其中考虑了浅层神经网络可达到的最低能量与更一般的函数之间的能量间隙。第二个是Allen-Cahn方程，其中分析了物理启发的神经网络的解决策略。在第二个问题中，与固定网格有限元方法相比，神经网络对低维移动界面的自适应性起着关键作用。理论结果旨在更好地理解两个基本挑战。首先是神经网络的自适应能否被用于空间非齐次变分问题的数值逼近。第二种是试图理解神经网络求解器不会优于传统求解器的精确情况。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。