The episode describes the Continuous Flow Operator (CFO), a novel neural framework for learning the continuous-time dynamics of Partial Differential Equations (PDEs), aimed at overcoming limitations found in conventional models like autoregressive schemes and Neural Ordinary Differential Equations (ODEs). CFO's key innovation is the use of a flow matching objective to directly learn the right-hand side of the PDE dynamics, utilizing the analytic velocity derived from spline-based interpolants fit to trajectory data. This approach uniquely allows for training on irregular and subsampled time grids while enabling arbitrary temporal resolution during inference through standard ODE integration. Across four benchmarks (Lorenz, 1D Burgers, 2D diffusion-reaction, and 2D ...