Identification of Energies from Observations of Evolutions

在某些能量景观的梯度流动或临界点的准静态演化方面，对其最小化所驱动的演化的研究已成为近年来研究的热点。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，借鉴了物理学中更成熟和经典的模型。例如，从Reynolds（1987）、Vicsek等人（1995）和Cucker-Smale（2007）的开创性论文开始，出现了大量描述共识或意见形成的模型，将信息交换建模为活跃主体（粒子）之间的远程社会互动（力）。然而，对于这种现象的分析，但更重要的是，对于这种现象的可靠和现实的数值模拟，人们需要完全理解和确定控制能量。不幸的是，除了可以通过相当精确地测量控制力来校准模型的物理情况外，对于物理学中的一些相关宏观模型以及生物学和社会科学中的大多数模型，控制能量远不能精确确定。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与现实案例中的数据相匹配。然而，这种只针对定性描述的态度倾向于将这一领域的一些研究减少到美丽和数学上有趣的玩具箱，这可能与现实生活中的场景无关。我们也直接参与了其中的一些发展，到目前为止，我们认识到其中一些模型的实际适用性存在一些重大限制。这个项目的目的是解决这个困难的任务，即提供一个数学框架，以便从直接观察相应的随时间变化的演化获得的数据中可靠地识别控制能量。这是一种新的反问题，超越了传统上认为的问题，因为正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中特定模型的能量学习，以及连续介质力学中材料断裂开始和扩展模型的能量学习。

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.