Identification of Energies from Observations of Evolutions

在过去的几年里，关于某些能量景观最小化所驱动的演化的研究，就其梯度流动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依赖时间的现象，借鉴了物理学中更成熟和经典的模型。例如，从雷诺兹(1987)的开创性论文开始，Vicsek et。艾尔(1995)和Cucker-Smer(2007)，已经有大量描述共识或意见形成的模型，将信息交换建模为活动主体(粒子)之间的远程社会交互(力量)。然而，对于分析，更关键的是要对这种现象进行可靠和现实的数值模拟，前提是对支配能量有一个完全的理解和决心。遗憾的是，除了可以通过相当精确地测量控制力来对模型进行校准的物理情况外，对于物理学中的一些相关宏观模型以及生物和社会科学中的大多数模型，控制能量远未被精确确定。事实上，在这些研究中，支配能量经常被预先确定为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影响，例如某些模式的形成，但很少或根本没有努力匹配来自真实案例的数据。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到漂亮的、数学上有趣的玩具箱，这可能与现实生活中的情景没有什么关系。我们还直接参与了其中一些发展，到目前为止，我们认识到其中一些模式在现实适用性方面存在某些重大限制。该项目的目的是解决一项艰巨的任务，即提供一个数学框架，以便从通过直接观测相应的随时间变化的演化所获得的数据中可靠地确定支配能量。这是一种新的反问题，超越了传统上认为的反问题，因为前向映射是一个强烈的非线性演化，高度依赖于产生初始条件的概率度量。由于我们的目标是精确的定量分析，并且非常具体，我们将攻击非局部相互作用支配的社会动力学中的特定模型的能量的学习，以及材料断裂起始和扩展模型的连续介质力学中的能量学习。

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.