New Approach to Inverse Problems for Differential Equation Networks

本项目的主要目标是解决微分方程网络(DENS)的辨识、观测和控制等方面的一些重要问题。洞是边上定义了微分方程的度量图，其顶点上的匹配条件耦合在一起。洞穴在许多科学和工程问题中起着基础性的作用。在原子尺度上，物理学家研究量子点和量子波导。在纳米尺度上，化学工程师对纳米棒的机械性能感兴趣。在细观尺度上，由弦和梁模拟的纤维网络是组织工程的关键，而在宏观尺度上，涉及管、板和壳的网格结构出现在土木工程和机械工程中。巢的控制和逆理论构成了迅速发展的图分析领域的一个重要组成部分。然而，这些理论对于应用来说是极其重要的，还没有得到充分的发展。控制、观察和辨识问题几乎只针对树，即没有圈的图，尽管一般的洞可能包含圈，但更重要的是应用。该项目为研究生和本科生提供了极好的培训机会。在该项目中，首席调查员(PI)和他的合作者将开发一种统一的方法来控制、观察和识别一般洞穴。该方法基于逆理论中的边界控制(BC)方法，该方法利用动力系统的可观性和可辨识性之间的深层联系。该项目的主要成果是：(A)解Den的反问题的唯一性和稳定性条件以及辨识系统参数的有效算法；(B)由Den描述的动力系统的能控性和能观性条件以及相应的控制和观测的构造方法。其核心思想是以偏微分方程解在某些边界和内部顶点的方向导数的形式适当地选择观测，以保证相应系统的可观测性和可辨识性。对于一般的洞穴，我们将指定传感器和/或执行器应该位于的顶点集合。我们方法的另一个组成部分是最近由PI和他的合作者开发的一种新的有效的剥叶(LP)方法，用于求解树上的微分方程组的反问题。在当前的项目中，我们计划将该方法扩展到一般的带圈图。基于BC和LP方法，将开发解决一般洞穴逆问题和控制问题的数值算法，并将进行广泛的数值实验。该奖项反映了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。

The main goal of this project is to solve some important problems on identification, observation and control of differential equation networks (DENs). DENs are metric graphs with differential equations defined on the edges coupled by matching conditions at the vertices. DENs play a fundamental role in many problems of science and engineering. On the atomistic scale, physicists work with quantum dots and quantum waveguides. On the nanoscale, chemical engineers are interested in mechanical properties of nanorods. On the mesoscale, fiber networks modeled by strings and beams are crucial for tissue engineering, whereas on the macroscale grid structures involving pipes, plates and shells appear in civil and mechanical engineering. Control and inverse theories of DENs constitute an important part of the rapidly developing area of analysis on graphs. Being tremendously important for applications, these theories have not been, however, sufficiently developed. Control, observation and identification problems were studied almost exclusively for trees, i.e. graphs without cycles, although general DENs, which may contain cycles, are more important for applications. The project provides excellent training opportunities for graduate and undergraduate students.In this project, the principal investigator (PI) and his collaborators will develop a unified approach to control, observation, and identification of general DENs. This approach is based on the Boundary Control (BC) method in inverse theory, which uses deep connections between observability and identifiability of dynamical systems. The main results of the project will be: (a) uniqueness and stability conditions for solving inverse problems for DENs and effective algorithms for identifying the systems' parameters; and (b) controllability and observability conditions of the dynamical systems described by DENs and the methods for constructing the corresponding control and observation. The central idea is a proper choice of observations in the form of directional derivatives of the solutions to PDEs on graphs at some boundary and internal vertices to guarantee observability and identifiability of the corresponding systems. For general DENs, we will specify the sets of vertices where sensors and/or actuators should be situated. Another component of our approach is a new effective leaf-peeling (LP) method for solving inverse problems for differential equations on trees that has been recently developed by the PI and his collaborators. In the current project we plan to extend this method to general graphs with cycles. Based on the BC and LP methods, numerical algorithms for solving inverse and control problems for general DENs will be developed and extensive numerical experiments will be performed.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.