统计物理关心从微观构型的分布中涌现的宏观规律，机器学习关心如何对高维空间数据变量的联合概率和条件概率分布建模。这两个领域之间有很多共通之处。在这个由鄂维南，王磊和张潘所组织的小型研讨会中，来自不同领域的学者将共同探讨这个交叉领域的核心问题，基础理论与新方法。

会议时间：2021年3月31日

会议地点：中科院理论物理研究所南楼6楼会议室

吴昊, 同济大学数学学院

Exploring Energy Landscapes by Normalizing Flows

Computing equilibrium states in thermodynamic many-body systems, such as solvated proteins, is a long-standing challenge. Combining normalizing flows and statistical mechanics, we develop Boltzmann Generators (BGs). By using neural networks to learn a coordinate transformation of the complex configurational equilibrium distribution to a distribution that can be easily sampled, BGs provid a new statistical mechanics tool that performs orders of magnitude faster than standard simulation methods. BGs can also be combined with traditional sampling procedures, including MCMC and LD, which leads to better representational power and sampling efficiency. In addition, some problems and challenges for future development of BGs (from my personal perspective) will be discussed.

王闯，中科院自动化所 （非监督机器学习）

Stochastic gradient iterations of linear models in high-dimensions with recycled training data

Stochastic gradient descent (SGD) is a fundamental tool in machine learning models. In this work, we study the dynamics of the standard SGD for linear models, such as linear regression, Gaussian mixture with two symmetric clusters, principal component analysis, and generalized linear models, in the high-dimensional limit. In particular, we provide an analytical formula for the training loss and population loss in the z-domain when the number of training samples and the dimension of learnable parameters both go to infinity with a fixed ratio. The main limiting formula only requires the training data X and the labels y satisfies certain self-average properties and does not directly rely on the concrete data model between X and y. Therefore, the proposed analysis method potentially paves a way to analyze non-linear models with more complicated structured data.

周海军, 中科院理论物理研究所

Planted XORSAT as a machine-learning challenge

The planted p-spin interaction model is a paradigm of random-graph systems possessing both a ferromagnetic phase and a disordered phase with the latter splitting into many spin glass states at low temperatures. Conventional simulated annealing dynamics is easily blocked by these low-energy spin glass states. Here we demonstrate that, actually this planted system is exponentially dominated by a microcanonical polarized phase at intermediate energy densities. There is a discontinuous microcanonical spontaneous symmetry breaking transition from the paramagnetic phase to the microcanonical polarized phase. We propose an unsupervised learning problem on microcanonically sampled configurations for inferring the planted ground state. This may serve as a very hard benchmark problem for evaluating different algorithms.

蒋滢

软物质体系微结构转变研究的机器学习方法探讨

软物质材料体系中的“熵-焓”竞争导致其在介观尺度的微结构异常复杂，其微纳结构与材料的宏观性能紧密相关，因此，如何有效的精确识别材料的微纳结构，进而研究体系中微结构转变的物理条件与规律，对于软物质功能材料的设计和加工至关重要。我们以两个典型的软物质体系为例：1. 单链高分子的“温致”链构象转变；2. 玻璃体系中的Gardner相变；采用机器学习中的非监督和监督学习方法，分别研究目标体系中的平衡态以及非平衡态结构转变。通过非监督学习中的线性和非线性方法，可以正确区分Coil、Globule、Anti-Mackay和Mackay结构，同时，自动提取了各种结构的物理序参量，此外，采用监督学习和非监督学习相结合的混合神经网络方法用来确定高分子构象相转变的临界点。对于非平衡态相变体系-玻璃体系，有针对性的设计基于深度学习的神经网络，通过有限尺寸分析，不仅明确阐明体系中存在Gardner（二级）相变，而且获得了该相变的临界指标。研究表明，机器学习将来有望应用于软物质体系中针对平衡态和非平衡态行为的理论研究。

全海涛，北京大学物理学院

Experimental Test of the Differential Fluctuation Theorem and a Generalized Jarzynski Equality for Arbitrary Initial States

Nonequilibrium processes of small systems such as molecular machines are ubiquitous in biology,chemistry, and physics but are often challenging to comprehend. In the past two decades, several exact thermodynamic relations of nonequilibrium processes, collectively known as fluctuation theorems, have been discovered and provided critical insights. These fluctuation theorems are generalizations of the second law and can be unified by a differential fluctuation theorem. Here we perform the first experimental test of the differential fluctuation theorem using an optically levitated nanosphere in both underdamped and overdamped regimes and in both spatial and velocity spaces. We also test several theorems that can be obtained from it directly, including a generalized Jarzynski equality that is valid for arbitrary initial states, and the Hummer-Szabo relation. Our study experimentally verifies these fundamental theorems and initiates the experimental study of stochastic energetics with the instantaneous velocity measurement.

Ref: Phys. Rev. Lett. 120, 080602 (2018)

庞龙刚 华中师大

探索深度学习在核物质相变与核结构反问题中的应用

通过高能核碰撞探索核物质相变与初态核结构是典型的反问题。目前实验正在进行不同能量下原子核碰撞的束流能量扫描工程，寻找核物质相变的临界点。此临界点被认为是量子色动力学相图上平滑过渡区（发生在宇宙早期和高能核碰撞）和可能出现的一阶相变区（发生在中低能核碰撞过程中）的分界点。实验上只能探测核碰撞产生的大量末态粒子在动量空间的分布，而无法直接看到中间过程中新的核物质形态 -- 夸克胶子等离子体 QGP 的产生和演化。分子动力学或相对论流体力学等蒙特卡洛仿真工具，可帮助研究初态核结构以及核物质相变种类如何影响 QGP 演化以及末态强子分布。但反过来，通过末态粒子分布，反推初态核结构以及核物质相变种类非常困难。我们使用卷积神经网络 CNN 证实核物质相变的信息在高能核碰撞复杂的动力学演化过程中幸存下来，并编码在末态强子分布中，CNN 可以帮助解码此信息。此外，我们还探索了点云神经网络以及神经网络的可解释性在核物质相变与核结构反问题中的应用。使用多体量子力学中的从头算方法研究重原子核结构比研究原子的电结构更加复杂，因为核子-核子相互作用势能除了类比于库伦势的 Yukawa 势，还有无法忽略的自旋轨道耦合、同位旋效应、张量势与三核子相互作用势能等。如何将基于神经网络的变分波函数（如 Fermi-Net）， 扩展到重原子核结构的计算，也是领域感兴趣的方向。

张林峰 北京大数据研究院

统计物理与机器学习的结合：从序参量选择引发的思考

我将介绍几个与合作者在统计物理与机器学习结合方面的研究，并把它们全都放在增强采样的框架内。由此我希望探讨其中我遇到的最困难的问题，即序参量的选择或学习。

黄海平 中山大学物理学院

神经网络的统计物理

该报告简要介绍课题组近三年来在神经网络的统计物理研究方面的进展，包括无监督学习的对称性破缺，深度网络的学习与退相关机制，以及深度学习的系综理论/算法。在报告最后将梳理一些开放问题以启发后续的理论研究。

汤雷翰 北京计算科学研究中心/香港浸会大学

The ultimate loss function — discovering new physics

Model development is a powerful methodology that allows integration of fundamental physical principles with emerging data and observations. In the language of machine learning (ML), this means that we need to combine the general learning algorithms with rule-based modelling to extract maximum understanding and predictive power under limited data. Currently, the two seldomly meet in the middle to enable iterative improvement of the research cycle. In this short presentation, I will use two examples, i.e., phase transitions in the 2D Ising and Potts model, and the traditional chinese medicine prescriptions, to illustrate challenges in pushing ML algorithms to discover fundamental features in the data, even when one is not aware of their existence.

张潘 中科院理论物理所

变分统计力学：平均场与神经网络

在报告中，张潘将介绍求解统计力学问题的变分方法：平均场方法与基于神经网络的变分自由能优化方法，并讨论神经网络方法相对于平均场、MCMC方法、张量网络方法的优势，劣势，以及解决真正困难问题需要面对的挑战。

王磊 中科院物理所

Two lessons from deep learning

1. Differentiable programming: one composes differentiable components into a computer program, then tune its parameters via gradient optimization. 2. Representation learning: one thinks neural networks for transforming representations of data, rather than fitting. I will explain these two lessons with examples from many-body physics.