  Data-driven modeling has become a key building block in computational science
and engineering. However, data that are available in science and engineering
are typically scarce, often polluted with noise and affected by measurement
errors and other perturbations, which makes learning the dynamics of systems
challenging. In this work, we propose to combine data-driven modeling via
operator inference with the dynamic training via roll outs of neural ordinary
differential equations. Operator inference with roll outs inherits
interpretability, scalability, and structure preservation of traditional
operator inference while leveraging the dynamic training via roll outs over
multiple time steps to increase stability and robustness for learning from
low-quality and noisy data. Numerical experiments with data describing shallow
water waves and surface quasi-geostrophic dynamics demonstrate that operator
inference with roll outs provides predictive models from training trajectories
even if data are sampled sparsely in time and polluted with noise of up to 10%.
