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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 Schrödinger 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 Schrödinger equation for atoms and molecules within and beyond the Born--Oppenheimer approximation.