发表机构
Institute of Theoretical & Computational Physics, University of Technology, Graz; Courant Institute, New York University; Division of Applied Mathematics, Brown University（格拉茨工业大学理论与计算物理研究所; 纽约大学柯朗数学科学研究所; 布朗大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出幽灵任务分配方法，可将非参数化系统有效参数化，用于构建参数化高斯过程，在数据有限的逆问题场景中表现优异，通过实验验证其效能并提供相关计算机代数程序的语法说明。
AI 中文摘要
近年来，物理信息机器学习受到了广泛关注。在数据有限的场景中，参数化高斯过程已成为热门方法。然而，现有方法常存在局限性，例如需要可参数化（也称为可控）系统或大量输出任务。本研究引入了一种名为“幽灵任务分配”的系统流程，利用辅助任务规避这些局限性。我们证明，此类幽灵任务可使任何非参数化系统实现有效参数化，从而能够算法化构建参数化高斯过程，同时保持所需任务数量（即输出维度）和潜函数数量处于较低水平。我们发现，幽灵任务分配在逆问题场景中表现尤为出色，即便在可用数据极少的情况下亦是如此。我们通过三项实验展示了幽灵任务分配的用法与效能，并与目前唯一适用于所有实验的其他方法进行了系统比较。我们为两个计算机代数程序提供了必要的语法说明，这两个程序可计算具有多项式或有理系数的系统的参数化。我们的理论结果可扩展至具有亚纯函数的系统。
英文摘要
Physics-informed machine learning has gained significant attention in recent years. In regimes of limited data, parametrized Gaussian processes have become popular. Existing approaches, however, often face limitations, such as requiring parametrizable (also called controllable) systems or a large number of output tasks. In this work, we introduce a systematic procedure we call "ghost tasking", using auxiliary tasks to circumvent these limitations. We prove that such ghost tasks can render any non-parametrizable system effectively parametrizable, enabling algorithmic construction of parametrized Gaussian Processes while keeping the number of required tasks (i.e. output dimensions) and latent functions low. We find that ghost tasking performs especially well in an inverse problem setting, even with very few available data. We show the usage and power of ghost tasking in three experiments, providing systematic comparisons to the only other currently available method applicable to all experiments. We provide necessary syntax and explications for two computer algebra programs that compute parametrizations for systems with polynomial or rational coefficients. Our theoretical results extend to systems with meromorphic functions.
Comments46 pages, 14 figures, for reproducibility: https://github.com/moserjo/GhostTask
