Title: Learning Reduced Fluid Dynamics. Abstract: Predicting the state evolution of ultra high-dimensional, time-reversible fluid dynamic system is a crucial but computationally expensive task. Model-reduction has been proven to be an effective method to reduce the computational cost by learning a low-dimensional state embedding. However, existing reduced models are irrespective of either the time reversible property or the nonlinear dynamics, leading to sub-optimal performance. We propose a model-based approach to identify locally optimal, model-reduced, time reversible, nonlinear fluid dynamic systems. Our main idea is to use stochastic Riemann optimization to obtain a high-quality a reduced fluid model by minimizing the expected trajectory-wise model-reduction error over a given distribution of initial conditions. To this end, our method formulates the reduced fluid dynamics as an invertible state transfer function parameterized by the reduced subspace. We further show that the reduced trajectories are differentiable with respect to the subspace bases over the entire Grassmannian manifold, under proper choices of timestep sizes and numerical integrators. Finally, we propose a loss function measuring the trajectory-wise discrepancy between the original and reduced models. By tensor precomputation, we show that gradient information of such loss functions can be evaluated efficiently over a long trajectory without time-integrating the high-dimensional dynamic system. Through evaluations on a row of simulation benchmarks, we show that our method lower the discrepancy by 45%-97% over conventional reduced models.