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

基于第一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。它最重要的应用之一是偏微分方程组(PDE)，连同它们的分析和数值算法，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，来自物理、生物和化学的日益复杂的数学模型对基于第一原理的方法解决实际问题的有效性提出了挑战，例如在流体湍流、分子动力学和大规模反问题中的方法。数值算法的一个主要障碍是所谓的维度诅咒。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诅咒方面表现出色，并在解决科学和工程中的复杂问题方面显示出巨大的潜力。这个项目旨在研究问题中的数学结构如何为创新的DNN的设计和分析提供信息，特别是在逆问题的背景下，其中未知参数是通过测量推断出来的，例如电阻抗断层扫描。此外，该项目中的编程组件将专注于培训下一代计算数学家。深度学习中的算子学习(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。这个项目将探索OPL的潜力，以缓解许多逆问题的不适定性，因为它强大的逼近能力与离线训练和在线预测特性相结合，可以实现高质量、快速的重建。该项目寻求通过将经典问题解决方法的数学结构集成到DNN体系结构中来连接OPL和经典方法。特别是，该项目将阐明注意机制的数学特性，它是最先进的DNN转换器的主干，例如GPT和AlphaFold 2中的那些。此外，该项目还将研究注意神经结构的灵活性，使注意机制能够与应用数学中的重要方法，如Galerkin投影或Fredholm型积分方程式，根据问题的先验数学结构相融合。这个项目还将通过希尔伯特空间中频谱理论的透镜深入研究关注的数学基础，试图理解象征性的查询-关键-值架构如何有助于变压器的丰富的代表性能力和多样化的近似能力。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。

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.