Collaborative Research: CCF: AF: Medium: Validated Soft Approaches to Parametric ODE Solving

协作研究：CCF：AF：中：经过验证的参数 ODE 求解软方法

• 批准号: 2212462
• 负责人: Chee Yap
• 金额: $ 42.89万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2212462&HistoricalAwards=false
• 关键词: Collaborative; Research; CCF; AF; Medium

Many physical, biological, and social processes are modeled as one ormore ordinary differential equations (ODEs) with unknown parameters.Usually there are three fundamental tasks in working with such ODEs: (1)checking whether the structure of the ODE even allows these parameters tobe estimated in principle, (2) if it does, numerically estimating theparameters, and (3) solving ODEs with the estimated values of theparameters. Thus, it is crucial to develop tools for the three tasks. Dueto its importance, there has been extensive research on developingnecessary mathematical theories, algorithms and software tools, withtremendous progress/achievements. Broadly, there have been two differentapproaches: symbolic and numeric, each with its own objective, theory,algorithms, and software tools. Roughly put, the symbolic approachesprioritize correctness over efficiency, while the numeric approachesprioritize efficiency over correctness. Naturally, they developed (oftendramatically) different sets of theories and algorithms. Consequently,there are currently two kinds of software tools: one correct but ofteninefficient, the other efficient but often incorrect. Hence, there is anutmost need and thus a challenge: develop a new approach (theory,algorithms) that can yield software tools that are both efficient andcorrect. In this project, the investigators propose a novel approach that has apotential to meet the challenges of efficiency and correctness forparametric ODEs. The approach may be described by the key phrase``validated and soft approach''. One may try to develop validated(correct) algorithms in two ways. (1) Use a symbolic approach. It alwaysproduces correct output, but is inefficient. (2) Use a numerical intervalapproach with modified notion of correctness, e.g., specifying a priorierror bounds. This allows the use of approximate arithmetic, providingefficiency, but this is only true for non-singular ODEs. For singularproblems, there is an implicit ``Zero Problem'' that does not yield tonumerical approximations, and may not even be Turing-computable. The softapproach overcomes this limitation by allowing indeterminacy for certaininputs: informally, inputs on the verge of singularity are allowed to haveindeterminate outputs. The resulting soft formulations of the problemsallow one to exploit and combine strengths of both symbolic and numericapproaches, resulting in algorithms that are correct (in the modifiedsense) and practical (efficient). The investigators' preliminary researchindicates that the validated soft approach is quite promising.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.

许多物理、生物和社会过程都被建模为一个或多个参数未知的常微分方程 (ODE)。使用此类 ODE 时通常需要完成三个基本任务：(1) 检查 ODE 的结构是否允许这些参数原则上估计，(2) 如果估计，则对参数进行数值估计，(3) 用参数的估计值求解 ODE。因此，开发完成这三项任务的工具至关重要。由于其重要性，人们在开发必要的数学理论、算法和软件工具方面进行了广泛的研究，并取得了巨大的进展/成就。从广义上讲，有两种不同的方法：符号方法和数字方法，每种方法都有自己的目标、理论、算法和软件工具。粗略地说，符号方法优先考虑正确性而不是效率，而数字方法则优先考虑效率而不是正确性。自然地，他们（通常戏剧性地）开发了不同的理论和算法。因此，目前有两种软件工具：一种是正确的，但通常效率低下，另一种是有效的，但通常是错误的。因此，最需要的也是一个挑战：开发一种新的方法（理论、算法），可以产生既高效又正确的软件工具。 在这个项目中，研究人员提出了一种新方法，有可能应对参数常微分方程的效率和正确性挑战。该方法可以用关键词“经过验证的软方法”来描述。人们可以尝试通过两种方式开发经过验证的（正确的）算法。 (1)使用象征性的方法。它总是产生正确的输出，但效率低下。 (2) 使用经过修改的正确性概念的数值区间方法，例如指定先验误差范围。这允许使用近似算术，从而提高效率，但这仅适用于非奇异 ODE。对于奇异问题，存在一个隐含的“零问题”，它不会产生数值近似，甚至可能无法进行图灵计算。软方法通过允许某些输入的不确定性克服了这一限制：非正式地，允许奇点边缘的输入具有不确定的输出。由此产生的问题的软表述允许人们利用和结合符号和数值方法的优势，从而产生正确的（在修改意义上）和实用的（有效的）算法。研究人员的初步研究表明，经过验证的软方法非常有前途。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。