  The use of implicit time-stepping schemes for the numerical approximation of
solutions to stiff nonlinear time-evolution equations brings well-known
advantages including, typically, better stability behaviour and corresponding
support of larger time steps, and better structure preservation properties.
However, this comes at the price of having to solve a nonlinear equation at
every time step of the numerical scheme. In this work, we propose a novel
operator learning based hybrid Newton's method to accelerate this solution of
the nonlinear time step system for stiff time-evolution nonlinear equations. We
propose a targeted learning strategy which facilitates robust unsupervised
learning in an offline phase and provides a highly efficient initialisation for
the Newton iteration leading to consistent acceleration of Newton's method. A
quantifiable rate of improvement in Newton's method achieved by improved
initialisation is provided and we analyse the upper bound of the generalisation
error of our unsupervised learning strategy. These theoretical results are
supported by extensive numerical results, demonstrating the efficiency of our
proposed neural hybrid solver both in one- and two-dimensional cases.
