Vector field divergence of predictive model output as indication of phase transitions

We introduce an alternative method to identify phase boundaries in physical systems. It is based on training a predictive model such as a neural network to infer a physical system's parameters from its state. The deviation of the inferred parameters from the underlying correct parameters will be most susceptible and diverge maximally in the vicinity of phase boundaries. Therefore, peaks in the vector field divergence of the model's predictions are used as indication of phase transitions. Our method is applicable for phase diagrams of arbitrary parameter dimension and without prior information about the phases. Application to both the two-dimensional Ising model and the dissipative Kuramoto-Hopf model show promising results.

Figure 1

Illustration of the divergence-based model to find phase transitions. (a) Parameters $p_0$ are uniformly drawn from parameter space to calculate corresponding samples $S$. A predictive model is trained to recognize the parameters for a given sample, such that the deviation $\delta p$ of the resulting prediction $p_{\mathrm{pred}}$ from the undertyling true $p_0$ is minimized on average. If the model can distinguish parameters between different phases better than within a phase, then crossing the border between different phases results in diverging predictions $\delta p\left(p_0\right)$, which provides a signature of a phase transition. Panel (b) illustrates this phenomenon by showing the vector field $\delta p\left(p_0\right)$ in the presence of two phases (blue and red) for the limit of zero resolution within a phase, where all predictions within a phase will be placed at its center of mass. The analytical argument leading to Eq. (4) for the opposite limiting case of high resolution is illustrated in panel (c), where the black bars correspond to the range of possible estimates of the order parameter at points on opposite sites of a phase transition, and the colored bars indicate the overlap to the neighboring bars. In this limit, the prediction will deviate toward the direction of higher overlap.

Figure 2

(a) Ising model on a two-dimensional lattice with coupling $J_x$ in horizontal direction (red) and coupling $J_y$ in vertical direction (blue). For large-enough coupling strength as compared to the temperature, the system will enter an ordered phase, whose spin configuration depends on the sign of couplings. The four different ordered phases are displayed with $J_x>0$ on the right, $J_x<0$ on the left, $J_y>0$ on top, and $J_y<0$ at the bottom, cf. Eq. (6). (b) Kuramoto-Hopf model with nearest-neighbor (green) and next-to-nearest neighbor (blue) couplings of the $i\mathrm{th}$ oscillator (red) on a two-dimensional array. While $V_1$ and $C$ couple only nearest neighbors, the parameter $V_2$ includes coupling to the next-to-nearest neighbors. (c) The patterns (I) and (II) are samples of stationary phases for different values of $V_2$ on a $128\times 128$ array with random initial conditions for constant $C=1$ and $V_1=5$, cf. Eq. (7). Panel (I) shows the spiral structures for the case $V_2/C=0.1$. For $V_2/C\approx 4$ several stable $\pi$ defects emerge in panel (II).

Figure 3

Panel (a) is a plot of the loss function $L$ from Eq. (1) as a function of the number of training epochs for both Ising and Kuramoto-Hopf model. The resulting deviation of the predictions (blue) and their divergence (red) from (2) for the coupling constant $J$ of the isotropic Ising model are plotted in panel (b) after one learning epoch and panel (c) after 60 epochs. The dashed black line indicates the analytical value $J/T=0.44$ of the phase transition. The error bars are obtained by splitting a total of 442 samples per point into 7 independent data sets. Each data set is in turn split in half into a training and an evaluation set. For the anisotropic Ising model, panel (d) shows the scaled deviations $\delta J$ (white arrows) after the first training epoch and the resulting divergence as a function of $J_x$ and $J_y$. The dashed yellow lines again indicate the Onsager result of the phase transition. Here a total of 100 samples per point were split into 10 independent data sets. Panels (e) and (f) show the network's predictions (black), the resulting deviation of the predictions (blue) and their divergence (red) from (2) for the coupling constant $V_2/C$ of the Kuramoto-Hopf model after five learning epochs and after 60 epochs, respectively. For this case we used 66 samples per point that were split into three independent data sets. For all models, each data set was trained 50 times from scratch for further averaging.