小数据区域中多尺度动态的物理感知、深度概率建模
Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime
摘要：
基于数据发现高维动力系统的有效的粗粒度（CG）模型是计算物理的一个独特的挑战，特别是在多尺度问题的背景下。本文提供了一个概率的角度，同时确定预测，低维粗粒度（CG）变量以及它们的动态。我们利用深度神经网络的表达能力来表示CG进化规律的右侧。此外，我们演示了如何领域知识，是非常经常可用的形式的物理约束（如守恒定律）可以与虚拟可观的新概念。这些约束除了导致物理上真实的预测之外，还可以显着减少所需的训练数据量，从而减少所需的、计算昂贵的多尺度模拟（小数据机制）的量。所提出的状态空间模型使用概率推理工具进行训练，与其他几种技术相比，不需要规定的精细到粗糙（限制）投影，也不需要状态变量的时间导数。所采用的配方是能够量化的预测的不确定性，以及重建的完整，精细尺度系统，允许选择感兴趣的数量后验的演变。我们证明了所提出的框架在高维系统中的移动粒子的功效。
英文摘要：
The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a probabilistic perspective that simultaneously identifies predictive, lower-dimensional coarse-grained (CG) variables as well as their dynamics. We make use of the expressive ability of deep neural networks in order to represent the right-hand side of the CG evolution law. Furthermore, we demonstrate how domain knowledge that is very often available in the form of physical constraints (e.g. conservation laws) can be incorporated with the novel concept of virtual observables. Such constraints, apart from leading to physically realistic predictions, can significantly reduce the requisite amount of training data which enables reducing the amount of required, computationally expensive multiscale simulations (Small Data regime). The proposed state-space model is trained using probabilistic inference tools and, in contrast to several other techniques, does not require the prescription of a fine-to-coarse (restriction) projection nor time-derivatives of the state variables. The formulation adopted is capable of quantifying the predictive uncertainty as well as of reconstructing the evolution of the full, fine-scale system which allows to select the quantities of interest a posteriori. We demonstrate the efficacy of the proposed framework in a high-dimensional system of moving particles.
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