Neural Stochastic Differential Equations on Compact State Spaces: Theory, Methods, and Application to Suicide Risk Modeling

The paper introduces a new class of stochastic differential equations (SDEs) whose solutions are guaranteed to stay within a specified compact polyhedral state space, addressing key limitations of existing SDE models for irregular, noisy, and partially observed ecological momentary assessment (EMA) data. It demonstrates that traditional chain‑rule constructions fail both theoretically and empirically, derives necessary constraints on drift and diffusion terms, and presents a parameterization that transforms arbitrary dynamics into constraint‑satisfying SDEs. Experiments on several real EMA datasets, including a large suicide‑risk study, show that this approach improves forecasting and optimization compared to standard latent neural SDE baselines, thereby enabling more trustworthy continuous‑time models for clinical time series.