Collaborative Research: Theory and Applications of Structure-Conforming Deep Operator Learning

基于第一性原理的方法在许多工程和科学学科中取得了相当大的成功，包括流体和固体力学，电磁学等。其中最重要的应用是偏微分方程（PDE），它与它们的分析和数值算法相结合，代表了人类为理解物质世界而开发的一些最强大的工具。然而，越来越复杂的数学模型所产生的物理，生物学和化学的挑战的第一原理为基础的方法来解决实际问题，如在流体湍流，分子动力学和大规模的逆问题的有效性。数值算法的一个主要障碍是所谓的维数灾难。 在图形处理单元和张量处理单元通用计算的进步的推动下，深度神经网络（DNN）和深度学习方法在对抗维度灾难方面表现出色，并在解决科学和工程中的复杂问题方面表现出巨大的潜力。该项目旨在研究问题中的数学结构如何为创新DNN的设计和分析提供信息，特别是在从测量中推断未知参数的逆问题中，例如电阻抗断层扫描。此外，该项目的编程部分将专注于培养下一代计算数学家。深度学习中的运算符学习（OpL）框架为解决具有挑战性和潜在不适定的基于偏微分方程的问题提供了独特的视角。该项目将探索OpL减轻许多逆问题的不适定性的潜力，因为其强大的近似能力与离线训练和在线预测特性相结合，可以实现高质量的快速重建。该项目旨在通过将经典问题解决方法中的数学结构集成到DNN架构中，来弥合OpL和经典方法。特别是，该项目将揭示注意力机制的数学特性，这是最先进的DNN Transformers的骨干，例如GPT和AlphaFold 2中的那些。 此外，该项目将研究注意力神经结构的灵活性，使注意力机制与应用数学中的重要方法融合，如Galerkin投影或Fredholm积分方程，根据问题的先验数学结构。这个项目还将通过希尔伯特空间中光谱理论的透镜深入研究注意力的数学基础，试图了解象征性的查询键价值体系结构为《变形金刚》提供了丰富的代表性和多样的近似能力。该奖项反映了NSF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评审进行评估，被认为值得支持的搜索.

The first-principle-based approach has achieved considerable success in numerous engineering and scientific disciplines, including fluid and solid mechanics, electromagnetism, and more. Among its most significant applications are partial differential equations (PDEs) which, in conjunction with their analysis and numerical algorithms, represent some of the most powerful tools humanity has ever developed for understanding the material world. However, increasingly complex mathematical models arising from physics, biology, and chemistry challenge the efficacy of first-principle-based approaches for solving practical problems, such as those in fluid turbulence, molecular dynamics, and large-scale inverse problems. A major obstacle for numerical algorithms is the so-called curse of dimensionality. Fueled by advances in Graphics Processing Unit and Tensor Processing Unit general-purpose computing, deep neural networks (DNNs) and deep learning approaches excel in combating the curse of dimensionality and demonstrate immense potential for solving complex problems in science and engineering. This project aims to investigate how mathematical structures within a problem can inform the design and analysis of innovative DNNs, particularly in the context of inverse problems where unknown parameters are inferred from measurements, such as electrical impedance tomography. Additionally, the programming component in this project will focus on training the next generation of computational mathematicians.The Operator Learning (OpL) framework in deep learning provides a unique perspective for tackling challenging and potentially ill-posed PDE-based problems. This project will explore the potential of OpL to mitigate the ill-posedness of many inverse problems, as its powerful approximation capability combined with offline training and online prediction properties lead to high-quality, rapid reconstructions. The project seeks to bridge OpL and classical methodologies by integrating mathematical structures from classical problem-solving approaches into DNN architectures. In particular, the project will shed light on the mathematical properties of the attention mechanism, the backbone of state-of-the-art DNN Transformers, such as those in GPT and AlphaFold 2. Furthermore, the project will examine the flexibility of attention neural architectures, enabling the fusion of attention mechanisms with important methodologies in applied mathematics, such as Galerkin projection or Fredholm integral equations, in accordance with the a priori mathematical structure of a problem. This project will also delve into the mathematical foundations of attention through the lens of spectral theory in Hilbert spaces, seeking to understand how the emblematic query-key-value architecture contributes to the rich representational power and diverse approximation capabilities of Transformers.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.