Unsupervised Manifold Clustering of Topological Phononics

Classification of topological phononics is challenging due to the lack of universal topological invariants and the randomness of structure patterns. Here, we show the unsupervised manifold learning for clustering topological phononics without any a priori knowledge, neither topological invariants nor supervised trainings, even when systems are imperfect or disordered. This is achieved by exploiting the real-space projection operator about finite phononic lattices to describe the correlation between oscillators. We exemplify the efficient unsupervised manifold clustering in typical phononic systems, including a one-dimensional Su-Schrieffer-Heeger–type phononic chain with random couplings, amorphous phononic topological insulators, higher-order phononic topological states, and a non-Hermitian phononic chain with random dissipations. The results would inspire more efforts on applications of unsupervised machine learning for topological phononic devices and beyond.

Figure 1

The unsupervised clustering of 1D phononic SSH model with projection operator $\widehat{P}$. (a) The SSH chain with finite size $L=20$, mass $m=1.0$, spring constant ${\kappa }_0=1.0$, and $\delta \in (0,1)$. The gray blocks indicate the unit cells. (b) The projection operator matrix $P_{ij}$ for the cases $\delta =0.2$ and $\delta =0.8$. (c) The first ten largest eigenvalues ${\lambda }_j$ with $ϵ=0.2$, $N=1000$, ${\omega }_c=\sqrt{{\kappa }_0/m}=1.0$. (d) Different topological classes are classified unsupervisedly, which coincides with the topological transition ($\delta =0.5$) in the phononic SSH model. After clustering, we label samples with different colors according to their $\delta$ values ($\delta >0.5$ or $\delta <0.5$) and confirm the classification. (e) Disordered phononic SSH chain with random elastic constants, where the bias $\mathrm{\Delta }\kappa \in (-0.75,0.75)$ and the spatially dependent random number ${\eta }_i\in (0,1)$. (f) The first ten largest eigenvalues ${\lambda }_j$ with $ϵ=1.0$, $N=1000$, ${\omega }_c=1.4$. (g) Different topological classes can still be well classified, even if the systems are disordered by noisy couplings. (h) The topological interface mode emerges between two random phononic chains with opposite signs of $\mathrm{\Delta }\kappa$ ($\mathrm{\Delta }\kappa =0.72$ and $\mathrm{\Delta }\kappa =-0.54$), and $L=40$. The field strength means the absolute amplitude of displacements (more details in the Supplemental Material [51]).

Figure 2

The unsupervised learning of amorphous topological phononics. (a) The demonstration of the amorphous structure, constructed by connecting adjacent centroids of Delaunay triangulation mesh [60]. The system has size $L=130\sim 150$. The projection operator is constructed in the coarse-grained space by averaging displacement field of oscillators in the dotted block of the original space. The reduced dimension is $L_r=10\times 10=100$. (b) The first ten largest eigenvalues ${\lambda }_j$ with $ϵ=0.2$, $N=300$, ${\omega }_c=1.0$. (c) Different classes that are classified unsupervisedly can coincide with different local Chern numbers $v$. The numbers of samples for $v=(0,-1,1)$ are 107, 93, 100, respectively. (d) The chiral edge modes in amorphous topological phononics with different local Chern number $v=\pm 1$. The amplitude and phase of phononic field are represented by the radius and colors (from blue to red) of the disks.

Figure 3

The unsupervised clustering of higher-order topological phononics. (a) The quadrupole higher-order topological models with coupling strength ${\kappa }_1={\kappa }_0(1-\delta )$ and ${\kappa }_2={\kappa }_0\delta$, $\delta \in (0,1)$. The dotted line means the negative counterpart. $L=8\times 8=64$. (b) The first ten largest eigenvalues ${\lambda }_j$ with $ϵ=1.0$, $N=500$, ${\omega }_c={\omega }_0$. (c) The structures are well classified into topological classes, corresponding to different value groups of $\delta$. (d) The energy levels as a function of $\delta$ show that the unsupervised learning can predict the higher-order topological transition, even at finite size. (Inset) Quadrupole corner states of the higher-order topological phononics.

Figure 4

The unsupervised clustering of 1D non-Hermitian phononic chain. (a) The $A/B$ oscillators have random dissipation losses, with alternatively large and small dissipation values ${\tau }_{A/B}^i={\tau }_0\pm {\eta }_i\mathrm{\Delta }\tau$, respectively. The length size $L=20$, the mass $m=1.0$, the viscosity ${\tau }_0=1.0$, the viscosity bias $\mathrm{\Delta }\tau \in (-0.5,0.5)$, and the random number ${\eta }_i\in (0,1)$. (b) The first ten largest eigenvalues ${\lambda }_j$ with $ϵ=1.0$, $N=500$, ${\omega }_c=1.35$. (c) Different classes are classified unsupervisedly, which coincides with the fact that the cases of ${\tau }_A>{\tau }_B$ ($\mathrm{\Delta }\tau >0$) and ${\tau }_A<{\tau }_B$ ($\mathrm{\Delta }\tau <0$) are topologically distinct. (d) The topological interface mode emerges between two non-Hermitian random phononic chains with opposite signs of $\mathrm{\Delta }\tau$ ($\mathrm{\Delta }\tau =0.4$ and $\mathrm{\Delta }\tau =-0.24$), and $L=60$.