Extrapolating Quantum Observables with Machine Learning: Inferring Multiple Phase Transitions from Properties of a Single Phase

We present a machine-learning method for predicting sharp transitions in a Hamiltonian phase diagram by extrapolating the properties of quantum systems. The method is based on Gaussian process regression with a combination of kernels chosen through an iterative procedure maximizing the predicting power of the kernels. The method is capable of extrapolating across the transition lines. The calculations within a given phase can be used to predict not only the closest sharp transition but also a transition removed from the available data by a separate phase. This makes the present method particularly valuable for searching phase transitions in the parts of the parameter space that cannot be probed experimentally or theoretically.

Figure 1

Schematic diagram of the kernel construction method employed to develop a Gaussian process model with extrapolation power. At each iteration, the kernel with the highest Bayesian information criterion (11) is selected. The labels in the boxes correspond to the kernel functions defined in (6)–(9).

Figure 2

Extrapolation of the polaron ground state momentum $K_{\mathrm{GS}}$ (left) and effective mass $m^{*}$ (right) across the sharp transition at ${\lambda }_{\mathrm{SSH}}=2{\alpha }^2/t\hslash \omega \approx 0.6$ for the model (2) with $\beta =0$. The black solid curves are the accurate quantum calculations. The symbols are the predictions of the GP models trained by the full polaron dispersions $E(K)$ at values of ${\lambda }_{\mathrm{SSH}}\le {\lambda }^{*}$, where ${\lambda }^{*}$ is shown by the vertical lines (solid for circles, dashed for triangles, and dot-dashed for pentagons). The GP models are used for interpolation (open symbols) and extrapolation (full symbols). The algorithm of Fig. 1 yields the kernel $k_{\mathrm{RQ}}\times k_{\mathrm{LIN}}+k_{\mathrm{RBF}}$ for the GP models represented by the triangles and pentagons and $k_{\mathrm{RQ}}\times k_{\mathrm{LIN}}\times k_{\mathrm{MAT}}$ for the circles. Left inset: The polaron dispersions used as input (dashed curves) and predicted by the GP model (solid curves) with ${\lambda }^{*}=0.5$ with the triangles showing the position of the dispersion minimum. Right inset: The polaron dispersions predicted by the GP model trained with ${\lambda }^{*}=0.6$ (solid curves) in comparison with the quantum calculations (symbols).

Figure 3

The polaron ground state momentum $K_{\mathrm{GS}}$ for the mixed model (2) as a function of $\beta /\alpha$ for ${\lambda }_{\mathrm{SSH}}=2{\alpha }^2/t\hslash \omega$. The color map is the prediction of the GP models. The curves are the quantum calculations from Ref. [37]. The models are trained by the polaron dispersions at the parameter values indicated by the white dots. No other information is used. The optimized kernel combination is $(k_{\mathrm{MAT}}+k_{\mathrm{RBF}})\times k_{\mathrm{LIN}}$ (upper panel) and $(k_{\mathrm{MAT}}\times k_{\mathrm{LIN}}+k_{\mathrm{RBF}})\times k_{\mathrm{LIN}}$ (lower panel).

Figure 4

GP prediction (solid curves) of the free energy density $f(T,m)$ of the mean-field Heisenberg model produced by Eq. (12) (dashed curves). Inset: The order parameter $m_0$ that minimizes $f(T,m)$: symbols, GP predictions; dashed curve, from Eq. (12). The GP models are trained with 330 points at $1.47<T<2.08$ (shaded area) and $-1.25<m<1.25$.