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 shallow water equations), the quintic CFO variant demonstrates superior long-horizon stability and significant data efficiency, often outperforming autoregressive baselines trained on complete datasets even when trained on only 25% of irregularly sampled data.

This episode provides a comprehensive monograph on diffusion models, detailing their foundational principles through three unifying perspectives: the Variational View (related to VAEs and DDPMs), the Score-Based View (rooted in EBMs and Score SDEs), and the Flow-Based View (connecting to Normalizing Flows and Flow Matching). The core concept involves defining a continuous forward process that adds noise and then learning a corresponding reverse process—a Stochastic Differential Equation (SDE) or Probability Flow Ordinary Differential Equation (PF-ODE)—to transform noise back into data. Much of the discussion focuses on the mathematical equivalence of these different formulations, the tractable training objectives (like Denoising Score Matching), and advanced techniques for accelerating the slow sampling process, including sophisticated numerical ODE solvers (like DPM-Solver) and distillation methods (such as Consistency Models). Finally, the monograph explores the theoretical connection between diffusion models and Optimal Transport (OT), suggesting that diffusion is related to, but not generally equivalent to, solving the optimal transport problem.

Reference:

Lai, C. H., Song, Y., Kim, D., Mitsufuji, Y., & Ermon, S. (2025). The Principles of Diffusion Models. arXiv preprint arXiv:2510.21890.

This episode introduces RainSeer, a novel, structure-aware framework for reconstructing high-resolution rainfall fields by treating radar reflectivity as a physically grounded structural prior. The authors argue that existing interpolation methods fail to capture localized extremes and sharp transitions crucial for applications like flood forecasting. RainSeer addresses two main challenges: the spatial resolution mismatch between volumetric radar scans and sparse ground-level station measurements (AWS), and the semantic misalignment caused by microphysical processes like melting and evaporation between the radar's view aloft and the rain that reaches the ground. The framework employs a Structure-to-Point Mapper for spatial alignment and a Geo-Aware Rain Decoder with a Causal Spatiotemporal Attention mechanism to model the physical transformation of hydrometeors during descent, demonstrating significant performance improvements over state-of-the-art baselines on two public datasets.