Training high-resolution AI-based Earth forecasting models is memory-intensive. Window-based Swin Transformers reduce the quadratic cost of global attention, but existing distributed systems such as AERIS primarily target pixel-level models and do not jointly support convolutional sampling modules and shifted-window execution. Long-lead rollout finetuning further increases activation memory. To address these challenges, we present TERRA, a hierarchical parallel training framework for high-resolution Earth forecasting. TERRA introduces Sampling-Aware Window, Sequence, and Tensor Parallelism (SAWSTP), which preserves spatially contiguous layouts for sampling modules and routes tokens into topology-aware ragged window layouts for Transformer execution. For long-lead finetuning, Memory Orchestration (MO) provides rollout-aware checkpoint planning and combines input buffering with budget-constrained activation offloading. Experiments on the $1/12^\circ$ GLORYS-based Wenhai workload show that TERRA supports models with up to 11.4B parameters on 96 H200 GPUs and sustains up to $39.76$ PFLOPS, achieving $65.0\%$ strong-scaling and $94.1\%$ weak-scaling efficiency. Compared with checkpoint-only policies, MO further reduces peak allocated GPU memory by $32.2\%$--$51.8\%$ with at most $20.0\%$ step-time overhead, which makes finetuning with smaller patch sizes and longer rollouts feasible for improved forecasting accuracy.
Neural stochastic differential equations (SDEs) have emerged as powerful tools for learning noisy or stochastic dynamics directly from data; however, existing approaches largely assume uncoupled and continuous noise, limiting their applicability to realistic stochastic drivers, and often scale poorly in time, requiring expensive autoregressive training. To address these limitations, we propose Neural Kolmogorov Equations (NKEs), a deterministic, infinite-dimensional reformulation of Neural SDEs based on the Kolmogorov Forward equation, transforming the learning problem from modelling individual stochastic trajectories to modelling the evolution of probability densities. NKEs learn general Lévy-type stochastic forcing directly through the operator structure of the KFE, and enable parallel-in-time training via a Lagrangian Galerkin projection and operator splitting. We evaluate NKEs on several stochastic benchmarks, including systems with coupled noise and jump processes, and verify that NKEs provide flexible models that accurately recover deterministic and stochastic dynamics with competitive predictive accuracy and improved training efficiency. Code and pretrained models will be released.