# Constraining the Dynamics of Deep Probabilistic Models

arXiv: https://arxiv.org/abs/1802.05680 · HTML: https://mlsanity.com/paper/1802.05680/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Feb 2018.

## Built on by (most cited first, 5 of 5)

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- [Compositional Modeling of Nonlinear Dynamical Systems with ODE-based Random Features](https://mlsanity.com/paper/2106.05960/) (2021)
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- [Deep Latent Force Models: ODE-based Process Convolutions for Bayesian Deep Learning](https://mlsanity.com/paper/2311.14828/) (2023)
- [Bayesian Nonparametric Mixed-Effect ODEs with Gaussian Processes](https://mlsanity.com/paper/2605.13088/) (2026)

## Abstract (arXiv)

We introduce a novel generative formulation of deep probabilistic models implementing "soft" constraints on their function dynamics. In particular, we develop a flexible methodological framework where the modeled functions and derivatives of a given order are subject to inequality or equality constraints. We then characterize the posterior distribution over model and constraint parameters through stochastic variational inference. As a result, the proposed approach allows for accurate and scalable uncertainty quantification on the predictions and on all parameters. We demonstrate the application of equality constraints in the challenging problem of parameter inference in ordinary differential equation models, while we showcase the application of inequality constraints on the problem of monotonic regression of count data. The proposed approach is extensively tested in several experimental settings, leading to highly competitive results in challenging modeling applications, while offering high expressiveness, flexibility and scalability.