Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration

We present a physical interpretation of machine learning functions, opening up the possibility to control properties of statistical systems via the inclusion of these functions in Hamiltonians. In particular, we include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results in the two-dimensional Ising model evidence that the field can induce an order-disorder phase transition by breaking or restoring the symmetry, in contrast with the field of the conventional order parameter which causes explicit symmetry breaking. The critical behavior is then studied by proposing a Hamiltonian-agnostic reweighting approach and forming a renormalization group mapping on quantities derived from the neural network. Accurate estimates of the critical point and of the critical exponents related to the operators that govern the divergence of the correlation length are provided. We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.

Figure 1

The architecture of the fully connected neural network. A renormalization group mapping is established between an original and a rescaled system based on the predictive function of the neural network.

Figure 2

Mean neural network function $\langle f\rangle$ versus external field $Y$ at inverse temperature $\beta =0.43,0,440687,0.45$ (right to left). The statistical uncertainty is comparable with the width of the lines.

Figure 3

Mean susceptibility of the neural network function $\langle {\chi }_f\rangle$ versus external field $Y$ at inverse temperature $\beta =0.43,0,440687,0.45$ (right to left). The statistical uncertainty is comparable with the width of the lines.

Figure 4

Mean neural network function $\langle f\rangle$ versus inverse temperature $\beta$ for an original and a rescaled system of size $L=L^{\prime }=32$.

Figure 5

Rescaled inverse temperature ${\beta }^{\prime }$ versus inverse temperature $\beta$. The intersection with $g\left(x\right)=x$ gives the critical fixed point $\beta ={\beta }^{\prime }={\beta }_c$. The dashed lines adjacent to the solid line indicate the statistical uncertainty.

Figure 6

Rescaled field $Y^{\prime }$ versus original field $Y$ at $\beta =0.440687$. The dashed lines indicate the statistical uncertainty.

Figure 7

An illustration of the blocking procedure with a rescaling factor of $b=2$ and the majority rule. For the two cases at the bottom, the choice of the rescaled degree of freedom was made arbitrarily.