  We study the dynamics of the Gaudin magnet ("central-spin model") using
machine-learning methods. This model is of practical importance, e.g., for
studying non-Markovian decoherence dynamics of a central spin interacting with
a large bath of environmental spins and for studies of nonequilibrium
superconductivity. The Gaudin magnet is also integrable, admitting many
conserved quantities: For $N$ spins, the model Hamiltonian can be written as
the sum of $N$ independent commuting operators. Despite this high degree of
symmetry, a general closed-form analytic solution for the dynamics of this
many-body problem remains elusive. Machine-learning methods may be well suited
to exploiting the high degree of symmetry in integrable problems, even when an
explicit analytic solution is not obvious. Motivated in part by this intuition,
we use a neural-network representation (restricted Boltzmann machine) for each
variational eigenstate of the model Hamiltonian. We then obtain accurate
representations of the ground state and of the low-lying excited states of the
Gaudin-magnet Hamiltonian through a variational Monte Carlo calculation. From
the low-lying eigenstates, we find the non-perturbative dynamic transverse spin
susceptibility, describing the linear response of a central spin to a
time-varying transverse magnetic field in the presence of a spin bath. Having
an efficient description of this susceptibility opens the door to improved
characterization and quantum control procedures for qubits interacting with an
environment of quantum two-level systems. These systems include electron-spin
and hole-spin qubits interacting with environmental nuclear spins via hyperfine
interactions or qubits with charge or flux degrees of freedom interacting with
coherent charge or paramagnetic impurities.
