Forecasting reservoir inflow is critical for making many policies, ranging from flood control and agriculture irrigation to water ecology management, hydropower generation, and landslide prevention. Prior studies mainly exploit autoregressive models -- e.g., recurrent neural networks (RNN) and its many variants -- to model the flow time series's temporal pattern. However, existing approaches rely on regular and accurate inflow observations, which either fail to predict multi-scale inflow (e.g., an hour, a day, or a month ahead prediction) or ignore the uncertainty of observations due to confounding factors such as snowmelt and precipitation. To address the limitations, we propose a novel inflow forecasting model by incorporating the uncertainty of the observations into the RNN model and the continuous-time dynamics of the latent states with neural ordinary differential equations (ODE). Our method, called FlowODE, explicitly encodes the stochasticity of hidden conditions in addition to the temporal dependencies among inflow observations. Moreover, FlowODE explores a continuum of layers instead of discrete RNNs to model the hidden states' dynamics, allowing us to infer the inflow at any time horizon flexibly. We conduct extensive experiments on the real-world datasets collected from two large-scale hydropower dams. The results show that our method consistently outperforms previous inflow forecasting models while providing adaptable predictions and a flexible balance between prediction accuracy and computational cost.