  Linear combinations of complex gaussian functions, where the linear and
nonlinear parameters are allowed to vary, are shown to provide an extremely
flexible and effective approach for solving the time-dependent Schr\"odinger
equation in one spatial dimension. The use of flexible basis sets has been
proven notoriously hard within the systematics of the Dirac--Frenkel
variational principle. In this work we present an alternative time-propagation
scheme that de-emphasizes optimal parameter evolution but directly targets
residual minimization via the method of Rothe's method, also called the method
of vertical time layers. We test the scheme using a simple model system
mimicking an atom subjected to an extreme laser pulse. Such a pulse produces
complicated ionization dynamics of the system. The scheme is shown to perform
very well on this model and notably does not rely on a computational grid. Only
a handful of gaussian functions are needed to achieve an accuracy on par with a
high-resolution, grid-based solver. This paves the way for accurate and
affordable solution of the time-dependent Schr\"odinger equation for atoms and
molecules within and beyond the Born--Oppenheimer approximation.
