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

arXiv: https://arxiv.org/abs/2508.17090 · HTML: https://mlsanity.com/paper/2508.17090/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 69 references that are in the map.
- First posted on arXiv 23 Aug 2025.

## Builds on (most cited first, 30 of 30)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
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- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Unsupervised Representation Learning for Time Series with Temporal Neighborhood Coding](https://mlsanity.com/paper/2106.00750/) (2021)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [Reflected Diffusion Models](https://mlsanity.com/paper/2304.04740/) (2023)
- [Learning Smooth Neural Functions via Lipschitz Regularization](https://mlsanity.com/paper/2202.08345/) (2022)
- [Continuously Differentiable Exponential Linear Units](https://mlsanity.com/paper/1704.07483/) (2017)
- [Diffusion Models for Constrained Domains](https://mlsanity.com/paper/2304.05364/) (2023)
- [Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations](https://mlsanity.com/paper/2102.06559/) (2021)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- [Neural Continuous-Discrete State Space Models for Irregularly-Sampled Time Series](https://mlsanity.com/paper/2301.11308/) (2023)
- [GLOBEM Dataset: Multi-Year Datasets for Longitudinal Human Behavior Modeling Generalization](https://mlsanity.com/paper/2211.02733/) (2022)
- [Metropolis Sampling for Constrained Diffusion Models](https://mlsanity.com/paper/2307.05439/) (2023)
- [Bayesian Structure Learning for Stationary Time Series](https://mlsanity.com/paper/1505.03131/) (2015)
- [Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions](https://mlsanity.com/paper/2410.03973/) (2024)
- [Enhancing Mixup-based Semi-Supervised Learning with Explicit Lipschitz Regularization](https://mlsanity.com/paper/2009.11416/) (2020)

## Abstract (arXiv)

Ecological Momentary Assessment (EMA) studies enable the collection of high-frequency self-reports of suicidal thoughts and behaviors (STBs) via smartphones. Latent stochastic differential equations (SDEs) are a promising model class for EMA data, as it is irregularly sampled, noisy, and partially observed. But SDE-based models suffer from two key limitations. (a) These models often violate domain constraints, undermining scientific validity and clinical trust of the model. (b) Training is numerically unstable without ad hoc fixes (e.g. oversimplified dynamics) that are ill-suited for high-stakes applications. Here, we develop a novel class of expressive SDEs whose solutions are provably confined to a prescribed compact polyhedral state space, matching the domains of EMA data. In this work, (1) we show why chain-rule based constructions of SDEs on compact domains fail, theoretically and empirically; (2) we derive constraints on drift and diffusion for general and stationary SDEs so their solutions remain in the desired state space; and (3), we introduce a parameterization that maps arbitrary (neural or expert-given) dynamics into constraint-satisfying SDEs. On several real EMA datasets, including a large suicide-risk study, our parameterization improves forecasts and optimization dynamics over standard latent neural SDE baselines. These contributions pave the way for principled, trustworthy continuous-time models of suicide risk and other clinical time series and extend applications of SDE-based methods (e.g. diffusion models) to domains with hard state constraints.