# Neural Stochastic Differential Equation

Spatial transcriptomics sequencing is destructive, so each time point yields unaligned cell snapshots and the intermediate trajectories and cross-scale intercellular interactions stay unobserved. stMIND decomposes each cell's expression drift into a cell-intrinsic program, a local graph-difference operator over relative expression differences and spatial geometry within a radius, and a mass-weighted nonlocal mean-field attention operator outside it, with neighborhoods co-evolving as cells migrate. On axolotl telencephalon regeneration and zebrafish embryogenesis benchmarks it reaches the highest gene-expression accuracy, anatomical consistency and spatial coherence; ablations show both operators are needed, and the learned fields locate a regenerative transition at 10–15 days post-injury.

Spatial transcriptomics sequencing is destructive, so each time point yields unaligned cell snapshots and the intermediate trajectories and cross-scale intercellular interactions stay unobserved. stMIND decomposes each cell's expression drift into a cell-intrinsic program, a local graph-difference operator over relative expression differences and spatial geometry within a radius, and a mass-weighted nonlocal mean-field attention operator outside it, with neighborhoods co-evolving as cells migrate. On axolotl telencephalon regeneration and zebrafish embryogenesis benchmarks it reaches the highest gene-expression accuracy, anatomical consistency and spatial coherence; ablations show both operators are needed, and the learned fields locate a regenerative transition at 10–15 days post-injury.

The authors introduce SDECast, a Neural Stochastic Differential Equation (SDE) framework for continuous-time probabilistic weather forecasting that extends SDE Matching to learn stochastic dynamics directly in physical space without repeated SDE simulation during training; on a simulated geophysical flow, SDECast recovers meaningful drift dynamics and faithfully reproduces the underlying continuous-time behavior, and it then scales to global weather forecasting at hourly resolution, producing skillful probabilistic forecasts for lead times up to five days.

The authors introduce SDECast, a Neural Stochastic Differential Equation (SDE) framework for continuous-time probabilistic weather forecasting that extends SDE Matching to learn stochastic dynamics directly in physical space without repeated SDE simulation during training; on a simulated geophysical flow, SDECast recovers meaningful drift dynamics and faithfully reproduces the underlying continuous-time behavior, and it then scales to global weather forecasting at hourly resolution, producing skillful probabilistic forecasts for lead times up to five days.

The authors propose HyperNSDE, a continuous-time generative model that encodes heterogeneous static covariates with an HI-VAE, maps the static latent representation through a hypernetwork to subject-specific drift parameters driving a latent Neural SDE, and jointly models observation times with a latent-state-dependent intensity process, trained via a deterministic–stochastic path decomposition with a signature-kernel objective; on simulated data and the VELOUR and PPMI cohorts it achieves better observation-time fidelity and several distributional and privacy metrics, while forecasting and correlation performance is mixed and affected by observation-grid regularity and trajectory smoothness.

The authors propose HyperNSDE, a continuous-time generative model that encodes heterogeneous static covariates with an HI-VAE, maps the static latent representation through a hypernetwork to subject-specific drift parameters driving a latent Neural SDE, and jointly models observation times with a latent-state-dependent intensity process, trained via a deterministic–stochastic path decomposition with a signature-kernel objective; on simulated data and the VELOUR and PPMI cohorts it achieves better observation-time fidelity and several distributional and privacy metrics, while forecasting and correlation performance is mixed and affected by observation-grid regularity and trajectory smoothness.