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物理学在机器学习中的应用以及两者的交叉融合正引起广泛关注, 尤其是在波动系统和扩散系统中. 本文重点关注波动与扩散物理系统和机器学习之间的内在联系以及对机器学习算法和物理实现的推进作用, 综述了波动系统和扩散系统中的机器学习研究, 介绍了部分最新研究成果. 文中首先讨论了监督学习的波动系统实现, 包括神经网络的波动光学实现、量子搜索的波动实现、基于波动系统的递归神经网络以及神经形态的非线性波动计算. 接着, 文中继续讨论了受扩散系统启发的机器学习算法, 如基于扩散动力学的分类算法, 基于热扩散的数据挖掘和信息过滤, 以及基于群体扩散的搜索优化等. 波动系统以其天然的并行性、高效、低能耗等优势, 通过丰富的波动力学和波动物理现象进行计算或算法模拟, 正成为机器学习的新型物理载体. 扩散系统中的物理机制可以启发构建高效的机器学习算法, 用于复杂系统和物理学研究中的分类、优化等问题. 期望通过对波动、扩散物理系统与机器学习内在联系的讨论, 能够为开发物理启发的新算法和硬件实现甚至软硬一体化带来抛砖引玉的启示.Recently, the application of physics to machine learning and the interdisciplinary convergence of the two have attracted wide attention. This paper focuses on exploring the internal relationship between physical systems and machine learning, and also on promoting machine learning algorithm and physical implementation. We summarize the researches of machine learning in wave systems and diffusion systems, and introduce some of the latest research results. We first discuss the realization of supervised learning for wave systems, including the wave optics realization of neural networks, the wave realization of quantum search, the recurrent neural networks based on wave systems, and the nonlinear wave computation of neural morphology. Then, we discuss the machine learning algorithms inspired by diffusion systems, such as the classification algorithm based on diffusion dynamics, data mining and information filtering based on thermal diffusion, searching for optimization based on population diffusion, etc. The physical mechanism of diffusion system can inspire the construction of efficient machine learning algorithms for the classification and optimization of complex systems and physics research, which may create a new vision for the development of physics inspired algorithms and hardware implementation, and even the integration of software and hardware.
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Keywords:
- wave systems /
- diffusion systems /
- machine learning /
- artificial neural network
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图 1 (a) 监督学习与无监督学习; (b) 一个单隐藏层神经网络的基本结构, 各层神经元根据权重系数相互连接; (c)利用无监督学习进行社交网络分析
Fig. 1. (a) Supervised learning and unsupervised learning; (b) the structure of a single hidden-layer neural network, where neurons are connected by weight coefficients; (c) unsupervised learning can be applied in social network analysis.
图 2 波在不同介质中的传播以及相应的线性矩阵运算 (a) 传统光学4f系统在傅里叶空间中实现乘法, 对应于在原空间中的卷积. 两个透镜分别实现傅里叶变换和傅里叶逆变换, 散射层对应于核矩阵, 场与薄散射层的相互作用相当于与对角矩阵的乘积; (b) 改进后的4f系统可以用光栅多次复制输入场, 并用不同的核矩阵进行卷积; (c) 通过类似4f的系统可以实现矢量-矩阵乘法, 实现一维行向量与稠密矩阵相乘, 得到一位列向量[6]
Fig. 2. Wave propagation through different media and the corresponding linear matrix operations: (a) A traditional optical 4f system realizes multiplication in Fourier space, which corresponds to the convolution in the original space; (b) modified 4f systems can copy the input field with a grating and use different kernels for convolution; (c) a 4f-type system can implement the vector–matrix multiplication[6].
图 3 (a) 基于相干光和全光学矩阵乘法的硅光子神经形态电路可用于实现元音分类[25]; (b) 基于波分复用(WDM)的分层结构构成的全光脉冲神经网络, 能够实现图像和语言识别[12]; (c) 改进的光学4f系统实现卷积神经网络, 提高图像分类性能[27]; (d) 全光衍射深度神经网络实现数字分类[29]; (e) 光学衍射元件与图像处理算法端对端协同设计[32]
Fig. 3. (a) Nanophotonic circuits based on coherent light and all-optical matrix multiplication is capable for vowel recognition[25]; (b) image and language recognition are achieved by an all-optical spiking neural networking with wavelength division multiplexing (WDM)[12]; (c) a design for an optical convolutional layer using a modified optical 4f system[27]; (d) an all-optical diffractive deep neural network that implements the digit classification[29]; (e) end-to-end learning paradigms of diffractive optics and processing algorithms[32].
图 5 标准RNN和波物理的对比 (a)具有离散的输入、输出序列的RNN的更新过程; (b) 具有连续输入、输出序列的波动系统系统的更新过程[41]
Fig. 5. Comparison of a standard RNN and a wave system: (a) The update process of an standard RNN with discrete input and output sequence; (b) the update process of a wave-based physical system with continuous input and output sequence[41].
图 6 神经形态的非线性波动网络 (a) 单波层前馈网络(SWFN)包含编码层、储蓄层和读出层, 其中波按照非线性偏微分方程演化; (b) 偏差
$ {\psi }_{b} $ 的演化、波的演化以及近似计算一维函数$ y={\rm{sin}}\left({\text{π}}x\right)/\left({\text{π}}x\right) $ 的结果; (c) SWFN用于学习鲍鱼数据集; (d) 用孤子训练通用的布尔逻辑门[46]Fig. 6. Neuromorphic computing by nonlinear waves: (a) Single wave-layer feed forward neural network (SWFN) with input layer, reservoir and readout layer, where the wave evolves according to a nonlinear partial differential equation; (b) the bias and wave evolution and results of learning the function
$ y={\rm{sin}}\left({\text{π}}x\right)/\left({\text{π}}x\right) $ ; (c) results of learning the abalone dataset; (d) training a universal logic gates by soliton gases[46].
图 7 利用扩散映射实现典型声子系统中的流形聚类 (a) 流形空间降维; (b) 流形空间的样本数据分布与势能绘景; (c) 流形空间的扩散与凝聚, 稳态显示出天然的聚类; (d)随机耦合的无序Su-Schrieffer-Heeger(SSH)声子链; (e) 无序非晶态声子的拓扑分类; (f) 一维非厄米声子链; (g) 高阶拓扑声子[21]
Fig. 7. Diffusion mapping in typical phononic systems to realize manifold clustering: (a) Dimension reduction in manifold space; (b) the probability distribution of samples and the effective landscape; (c) along with evolution, the samples diffuse and finally concentrate on positions with minimum local potentials, which indicates the clustering; (d)–(g) applications in disordered photonic SSH chain, amorphous topological phononics, 1D non-Hermitian phononic chain, high-order topological phononics[21].
图 8 基于扩散映射的无监督学习方法适用于解决不同物理系统中的拓扑分类问题 (a) 一维XY模型拓扑序的检测[52]; (b) 扩散映射能够不借助边缘态, 识别Haldane模型描述的拓扑相变点[54]; (c) 量子系统中的扩散映射算法, 能够无监督地识别
$ {\mathbb{Z}}_{3} $ 横场伊辛模型的量子相[55]Fig. 8. The unsupervised learning with diffusion map is applied to solve topology identification in different physical systems: (a) Identifying the topological order in 1-dimensional XY model[52]; (b) detection of the phase transition for the Haldane model without the edge states[54]; (c) diffusion maps in learning quantum phases with a
$ {\mathbb{Z}}_{3} $ transverse field Ising model[55].
图 9 标号图表明6个点(项目)之间的连接关系, 右侧是对应的权重矩阵和邻接矩阵
Fig. 9. Labelled graphs show the connection of 6 points (items), and the corresponding degree matrix and adjacency matrix are on the right side.
图 10 (a) 全局及局部粒子群优化算法示意图; (b) 粒子群优化算法中速度及位置更新示意图[68]; (c) 多目标优化的二维SnSe材料定向设计工作流程图; (d) 室温下(300 K), 二维SnSe材料单层结构的自由能; (e) 图(d) 中第一Pareto前沿(红线)上的四种新型单层结构的三视图, 深灰色和绿色的球分别表示Sn原子和Se原子[70]
Fig. 10. (a) The diagram of PSO; (b) The schematic diagram of the velocity and position updates in PSO[68]; (c) workflow of the multi-objective optimization for 2D SnSe materials design; (d) thermopower landscape at room temperature (300 K) versus the free energy of 2D SnSe materials; (e) four 2D SnSe structures on the first Pareto front, where the dark gray and green balls denote Sn and Se atoms, respectively[70].
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