  We introduce the group-equivariant autoencoder (GE-autoencoder) -- a deep
neural network (DNN) method that locates phase boundaries by determining which
symmetries of the Hamiltonian have spontaneously broken at each temperature. We
use group theory to deduce which symmetries of the system remain intact in all
phases, and then use this information to constrain the parameters of the
GE-autoencoder such that the encoder learns an order parameter invariant to
these ``never-broken'' symmetries. This procedure produces a dramatic reduction
in the number of free parameters such that the GE-autoencoder size is
independent of the system size. We include symmetry regularization terms in the
loss function of the GE-autoencoder so that the learned order parameter is also
equivariant to the remaining symmetries of the system. By examining the group
representation by which the learned order parameter transforms, we are then
able to extract information about the associated spontaneous symmetry breaking.
We test the GE-autoencoder on the 2D classical ferromagnetic and
antiferromagnetic Ising models, finding that the GE-autoencoder (1) accurately
determines which symmetries have spontaneously broken at each temperature; (2)
estimates the critical temperature in the thermodynamic limit with greater
accuracy, robustness, and time-efficiency than a symmetry-agnostic baseline
autoencoder; and (3) detects the presence of an external symmetry-breaking
magnetic field with greater sensitivity than the baseline method. Finally, we
describe various key implementation details, including a new method for
extracting the critical temperature estimate from trained autoencoders and
calculations of the DNN initialization and learning rate settings required for
fair model comparisons.
