This paper introduces NeuralGCM, a hybrid atmospheric model that integrates machine learning with traditional differentiable physics to improve global precipitation simulations. Unlike older models that rely on high-resolution simulations for training, this framework is trained directly on satellite observations, specifically the IMERG dataset. By leveraging this observational data, the model effectively corrects common biases in extreme weather events and the diurnal cycle of rainfall. In comparative tests, the model outperformed the ECMWF ensemble in mid-range forecasting and showed superior accuracy over CMIP6 climate models. Additionally, the architecture is exceptionally efficient, running simulations at speeds orders of magnitude faster than conventional general circulation models. These findings suggest that hybrid neural models offer a more reliable and computationally accessible path for predicting future climate impacts.

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.