Neural Networks for Stationary and Evolutionary Variational Problems

人工神经网络已经成为数据科学中的主要模型之一，应用于从图像分类到自然语言处理的各种应用中。他们在这些不同领域取得的经验成功激发了人们在数值分析和科学计算等新方向进一步应用这些模型的兴趣。这个项目的目的是加深对用于数值分析和科学计算的神经网络模型的能力和局限性的理解，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应用于工程领域的敏感领域是至关重要的。作为该项目的一部分，将对研究生进行培训，将数据科学和深度学习的现代工具纳入研究生课程，并计划开展推广活动，以吸引本科生以及STEM中未被充分代表的群体进入该研究领域。本工作的重点是在基于变分的模型中使用神经网络模型，以两个案例研究为目标。第一类是与呈现拉夫伦蒂耶夫能隙现象的泛函有关，其中考虑了浅层神经网络所能达到的最低能量与更一般函数之间的能隙。其次是Allen-Cahn方程，分析了物理启发神经网络的求解策略。在第二个问题中，与固定网格有限元方法相比，神经网络对低维运动界面的适应性起着关键作用。理论结果旨在更好地理解两个根本挑战。第一个问题是能否利用神经网络的自适应能力对空间上非常不齐次的变分问题进行数值逼近。第二个奖项旨在了解神经网络解算器不会超越传统解算器的确切情况。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.