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的辨识、观测和控制等方面的一些重要问题。DEN是度量图，其微分方程定义在边上，并通过顶点处的匹配条件耦合。DENs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，物理学家使用量子点和量子波导。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对于组织工程至关重要，而在宏观尺度上，涉及管、板和壳的网格结构出现在土木和机械工程中。DENS的控制与逆理论是图的分析领域的一个重要组成部分。由于这些理论对实际应用极为重要，因此还没有得到充分的发展。控制，观察和识别问题的研究几乎完全是针对树，即没有循环的图，虽然一般的DEN，其中可能包含循环，是更重要的应用程序。该项目为研究生和本科生提供了良好的培训机会。在该项目中，主要研究者（PI）及其合作者将开发一种统一的方法来控制，观察和识别一般DEN。该方法基于逆理论中的边界控制（BC）方法，该方法利用了动力系统的可观测性和可辨识性之间的深层联系。该项目的主要成果将是：（a）求解离散元反问题的唯一性和稳定性条件以及确定系统参数的有效算法;（B）离散元描述的动力系统的可控性和可观测性条件以及构造相应控制和观测的方法。其中心思想是以图上某些边界和内部顶点处的偏微分方程解的方向导数形式适当选择观测值，以保证相应系统的可观测性和可识别性。对于一般的DEN，我们将指定传感器和/或致动器应该位于的顶点的集合。我们的方法的另一个组成部分是一个新的有效的剥叶（LP）的方法来解决反问题的微分方程的树，最近开发的PI和他的合作者。在目前的项目中，我们计划将这种方法扩展到一般的图与循环。基于BC和LP方法，将开发用于解决一般DEN的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。

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.