CAREER: Towards a general recipe for fast high-dimensional scientific computing

这个项目解决了求解多体Fokker-Planck和Schrödinger偏微分方程组(PDE)中的维度诅咒，这是理解和预测分子结构、材料属性、化学反应、极端天气事件和各种物理尺度上的图案形成的基础。研究人员旨在通过开发一种结合张量网络、蒙特卡罗和凸优化方法优点的迭代求解器来开发一种通用的方法来克服这些偏微分方程组的高维问题。其主要思想是只求解少量的解的描述符(例如，解的统计矩)，然后通过这些描述符来恢复解的压缩表示。改进的解决程序将通过开发用于物理和化学模拟的新计算工具来促进材料基因组倡议。拟议的教育计划旨在培养数学和物理科学交叉领域的定量研究研究生，培养新一代研究人员，他们具备处理科学计算、机器学习和量子计算等国家关心的问题的能力。通过研究咨询、暑期辅导计划和发展量子力学暑期学校，该提案旨在鼓励未被充分代表的学生参与科学研究和高等教育。目前，蒙特卡罗方法由于其灵活性，已经在许多实际的物理和化学模拟工具中获得了广泛的成功，尽管存在混合时间慢和方差高等潜在缺点。另一种表征化学/物理系统的方法是确定地求解整个空间上的偏微分方程组的解，这只能适用于小的(从而是低维的)问题。拟议的研究试图通过开发迭代求解器来解决基于采样的方法和偏微分方程方法的局限性。它包括通过在短时蒙特卡罗模拟和估计张量网络ansatz之间交替进行快速迭代。它依靠基于凸优化的初始化策略来减少迭代次数。由于只需要估计少量的张量网络参数，因此大大降低了传统抽样方法的蒙特卡罗方差。另一方面，它允许随机操作，大大扩展了张量网络方法的灵活性。为此，提出了一种新的基于张量网络的生成模型，其中密度可以从经验样本中学习，而不需要使用任何优化。它将通过提供新的密度估计器和分析来影响统计，而不受维度诅咒的影响。在此基础上，提出了一种新的求解高维偏微分方程组的凸优化矩方法。这将把运筹学中传统使用的数学编程工具发展成为物理模拟的有效工具。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。

This project addresses the curse of dimensionality in solving the many-body Fokker-Planck and Schrödinger partial differential equations (PDEs), which are fundamental to understanding and predicting molecular structures, material properties, chemical reactions, extreme weather events, and pattern formations at various physical scales. The investigator aims to develop a general method to overcome high dimensionality of these PDEs via developing an iterative solver that combines the advantages of tensor-network, Monte Carlo, and convex optimization methods. The main idea is to solve for only a small number of descriptors of the solution (e.g., the statistical moments of the solution) and then recover a compressed representation of the solution through these descriptors. An improved solution procedure will contribute to the Material Genome Initiative by developing new computational tools for physical and chemical simulations. The proposed education plan aims to train graduate students for quantitative research at the intersection of mathematical and physical sciences, fostering a new generation of researchers well-equipped to tackle problems of national interest in scientific computing, machine learning, and quantum computing. Through research advising, a summer mentoring program, and the development of a summer school for quantum mechanics, the proposal aims to encourage the participation of underrepresented students in scientific research and higher education. Currently, Monte Carlo methods have been widely successful in many practical physical and chemical simulation tools due to their flexibility, despite potential drawbacks such as slow mixing time and high variance. An alternative approach to characterizing chemical/physical systems is to deterministically solve for the solution of a PDE over the entire space, which can work only for small (and consequently low-dimensional) problems. The proposed research attempts to address the limitations of sampling-based and PDE approaches through the development of an iterative solver. It involves fast iterations achieved by alternating between short-time Monte Carlo simulations and estimating a tensor-network ansatz. It relies on the initialization strategy underpinned by convex optimization in order to reduce the number of iterations. Monte Carlo variance of traditional sampling methods is significantly reduced since only a small number of tensor-network parameters need to be estimated. On the other hand, it significantly extends the flexibility of tensor-network methods by allowing stochastic operations. To this end, a novel tensor-network-based generative model is proposed where density can be learned from empirical samples without the use of any optimization. It would impact statistics by providing new density estimators and analysis without the curse of dimensionality. Further, a new moment method based on convex optimization is proposed for solving high-dimensional PDEs. This would develop mathematical programming, a tool traditionally used in operations research, into an effective tool for physics simulations.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.