Statistical Computing in Modern Scientific Analysis

现代科学分析使用越来越精确的仪器来记录越来越多的实验数据。虽然这对科学发现有很大的希望，但从复杂和相互作用的仪器误差中提取合理的结论对统计计算构成了巨大的挑战。本研究计划的目标是为自然科学中的各种动态过程开发可靠的测量模型，并开发用于科学探究的计算方法。本研究计划的中心焦点是微分方程（DEs）的统计推断。DE模型自然而忠实地捕捉了无数动态过程的底层物理。然而，这些模型的统计推断是非常计算密集的，并且容易受到大的数值舍入误差的影响。本研究计划将通过追求以下目标来解决这些问题：-目标1：通过将舍入误差直接纳入统计模型来正确考虑舍入误差。-目标2：用推断更直接的替代品有效地替代DE模型。-目标3：使用机器学习方法和工具包开发DEs的近似推理方法。实现这些目标的途径将在各种DE模型和其他孤立的计算推理策略之间提出重要的方法联系。通过机器学习方法解决DE推理问题将产生快速、灵活和可靠的计算工具，以促进DE在科学探究中的广泛使用，此外还为学生在加拿大劳动力中追求有影响力的数据科学职业提供优秀的培训。

Modern-day scientific analyses employ ever more accurate instruments to record ever larger amounts of experimental data. While this has great promise for scientific discovery, extracting sound conclusions for complex and interacting instrumental errors poses a formidable challenge in statistical computing. The goal of this research program is to develop reliable measurement models for various dynamical processes in the natural sciences, and to develop the computational methods to use these models for scientific inquiry. A central focus of this research program is statistical inference for differential equations (DEs). DE models naturally and faithfully capture the underlying physics of countless dynamical processes. However, statistical inference for these models is extremely computationally intensive, and susceptible to large numerical round-off errors. This research program will address these issues by pursuing the following objectives: - Objective 1: To correctly account for round-off errors by incorporating them directly into the statistical model. - Objective 2: To effectively substitute DE models by surrogates for which inference is much more straightforward. - Objective 3: To develop approximate methods of inference for DEs using machine learning methods and toolkits. The path towards realizing these objectives will propose important methodological connections between a variety of DE models and otherwise isolated computational inference strategies. Addressing DE inference via machine learning methods will produce fast, flexible and reliable computational tools to promote widespread use of DEs for scientific inquiry -- in addition to providing excellent training for students to pursue impactful data science careers in the Canadian workforce.