Identification of Energies from Observations of Evolutions

从进化观察中识别能量

• 批准号: 313937443
• 负责人: Professor Dr. Massimo Fornasier
• 金额: --
• 依托单位: Lehrstuhl für Angewandte Numerische Analysis und Optimierung und Data Analysis (M15)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/313937443?language=en
• 关键词: Identification; Energies; Observations; Evolutions

The study of evolutions driven by the minimization of certain energetic landscapes, in terms of their gradient flows or quasi-static evolutions of critical points, has been the subject of intensive research in the past years. Some of the most recent models are aiming at describing time-dependent phenomena also in biology or even in social dynamics, borrowing a leaf from more established and classical models in physics. For instance, starting with the seminal papers of Reynolds (1987), Vicsek et. al. (1995), and Cucker-Smale (2007), there has been a flood of models describing consensus or opinion formation, modeling the exchange of information as long-range social interactions (forces) between active agents (particles). However, for the analysis, but even more crucially for the reliable and realistic numerical simulation of such phenomena, one presupposes a complete understanding and determination of the governing energies. Unfortunately, except for physical situations where the calibration of the model can be done by measuring the governing forces rather precisely, for some relevant macroscopical models in physics and most of the models in biology and social sciences the governing energies are far from being precisely determined. In fact, very often in these studies the governing energies are just predetermined to be able to reproduce, at least approximately or qualitatively, some of the macroscopical effects of the observed dynamics, such as the formation of certain patterns, but there has been little or no effort of matching data from real-life cases. This attitude aiming just at a qualitative description tends however to reduce some of the investigations in this area to beautiful and mathematically interesting toy-cases, which have likely little to do with real-life scenarios. We also have been directly involved in some of these developments and we recognize by now certain significant limitations towards the realistic applicability of some of these models. The aim of this project is to approach the difficult task of providing a mathematical framework for the reliable identification of the governing energies from data obtained by direct observations of corresponding time-dependent evolutions. This is a new kind of inverse problem, beyond more traditionally considered ones, as the forward map is a strongly nonlinear evolution, highly dependent on the probability measure generating the initial conditions. As we aim at a precise quantitative analysis, and to be very concrete, we will attack the learning of the energies for specific models in social dynamics governed by nonlocal interactions and in continuum mechanics for models of material fracture initiation and propagation.

在过去的几年里，研究由某些能量景观的最小化驱动的演化，就其梯度流或临界点的准静态演化而言，一直是深入研究的主题。一些最新的模型旨在描述生物学甚至社会动力学中的时间依赖现象，从物理学中更成熟和经典的模型中借鉴。例如，从Reynolds（1987），Vicsek et.（1995）和Cucker-Smale（2007），已经有大量的模型描述共识或意见形成，将信息交换建模为主动代理（粒子）之间的远程社会交互（力）。然而，对于分析，但更重要的是，对于这种现象的可靠和现实的数值模拟，一个先决条件是完全理解和确定的支配能量。不幸的是，除了物理情况下，模型的校准可以通过测量的控制力相当精确地完成，在物理学和生物学和社会科学的大多数模型的一些相关的宏观模型的控制能量是远远不能被精确地确定。事实上，在这些研究中，控制能量通常只是预先确定的，以便能够至少近似地或定性地再现所观察到的动力学的一些宏观效应，例如某些模式的形成，但是很少或根本没有努力匹配来自现实生活案例的数据。然而，这种只着眼于定性描述的态度往往会将这一领域的一些研究减少为美丽的和数学上有趣的玩具案例，这些案例与现实生活中的场景可能没有什么关系。我们还直接参与了其中一些发展，我们现在认识到其中一些模型的现实适用性存在某些重大限制。这个项目的目的是接近困难的任务，提供一个数学框架，从直接观测相应的随时间变化的演变所获得的数据的可靠识别的管理能源。这是一种新的反问题，超越了传统上考虑的问题，因为正向映射是一个强非线性演化，高度依赖于生成初始条件的概率测度。由于我们的目标是精确的定量分析，并且非常具体，我们将攻击由非局部相互作用控制的社会动力学和连续介质力学中材料断裂起始和传播模型的特定模型的能量学习。