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

该项目解决了在求解多体Fokker-Planck和Schrödinger偏微分方程（PDEs）时的维度诅咒，这些偏微分方程是理解和预测分子结构、材料性质、化学反应、极端天气事件和各种物理尺度上的模式形成的基础。研究者的目标是通过开发一种结合张量网络、蒙特卡罗和凸优化方法的迭代求解器，开发一种通用方法来克服这些偏微分方程的高维性。其主要思想是只求解解的少量描述符（例如，解的统计矩），然后通过这些描述符恢复解的压缩表示。改进的解决程序将通过开发新的物理和化学模拟计算工具，为材料基因组计划做出贡献。拟议的教育计划旨在培养研究生在数学和物理科学交叉领域进行定量研究，培养新一代研究人员，他们有能力解决科学计算、机器学习和量子计算方面的国家利益问题。该提案旨在通过研究建议、暑期辅导计划和量子力学暑期学校的发展，鼓励代表性不足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，尽管存在诸如混合时间慢和高方差等潜在的缺点。表征化学/物理系统的另一种方法是在整个空间上确定地求解PDE的解，这种方法只能用于小的（因此是低维的）问题。提出的研究试图通过开发迭代求解器来解决基于抽样和PDE方法的局限性。它涉及通过在短时蒙特卡罗模拟和估计张量网络分析之间交替进行的快速迭代。它依赖于凸优化支持的初始化策略，以减少迭代次数。由于只需要估计少量张量网络参数，传统采样方法的蒙特卡罗方差显著减小。另一方面，它通过允许随机操作，极大地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，该模型可以从经验样本中学习密度，而无需使用任何优化。它将通过提供新的密度估计器和没有维度诅咒的分析来影响统计。在此基础上，提出了一种新的求解高维偏微分方程的凸优化矩法。这将发展数学规划，一种传统上用于运筹学的工具，成为物理模拟的有效工具。该奖项反映了美国国家科学基金会的法定使命，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.