Neural Networks for Stationary and Evolutionary Variational Problems

人工神经网络已成为数据科学中的主导模型之一，用于从图像分类到自然语言处理的应用。他们在这些不同领域的经验成功引发了人们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。该项目旨在更深入地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能够广泛部署在工程领域的敏感领域至关重要。研究生将作为该项目的一部分接受培训，数据科学和深度学习的现代工具将被纳入研究生课程，并计划开展外联活动，以吸引本科生以及STEM领域代表性不足的群体进入这一研究领域。这项工作的重点是将神经网络模型用于基于变分法的模型中使用的数值算法，针对两个案例研究。第一个是有关的泛函，表现出拉夫连季耶夫间隙现象，其中的最低能量之间的差距，可实现的浅神经网络和更一般的功能被认为是。第二个是Allen-Cahn方程，分析了物理启发的神经网络的解决方案。在第二个问题中，神经网络对低维移动界面的自适应性在与例如固定网格有限元方法相比时起着关键作用。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性是否可以利用空间非常不均匀的变分问题的数值逼近。该奖项反映了NSF的法定使命，并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。

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.