# Bayesian differential programming for robust systems identification under uncertainty

arXiv: https://arxiv.org/abs/2004.06843 · HTML: https://mlsanity.com/paper/2004.06843/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2020.

## Built on by (most cited first, 16 of 16)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data](https://mlsanity.com/paper/2103.03385/) (2021)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [PCE-PINNs: Physics-Informed Neural Networks for Uncertainty Propagation in Ocean Modeling](https://mlsanity.com/paper/2105.02939/) (2021)
- [Weak-Form Inference for Hybrid Dynamical Systems in Ecology](https://mlsanity.com/paper/2405.20591/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Joint Parameter and Parameterization Inference with Uncertainty Quantification through Differentiable Programming](https://mlsanity.com/paper/2403.02215/) (2024)
- [Multi-fidelity climate model parameterization for better generalization and extrapolation](https://mlsanity.com/paper/2309.10231/) (2023)
- [History-Based, Bayesian, Closure for Stochastic Parameterization: Application to Lorenz '96](https://mlsanity.com/paper/2210.14488/) (2022)
- [Sparse Identification of Nonlinear Dynamics with Conformal Prediction](https://mlsanity.com/paper/2507.11739/) (2025)
- [Scalable Bayesian optimization with high-dimensional outputs using randomized prior networks](https://mlsanity.com/paper/2302.07260/) (2023)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Towards Learning Stochastic Population Models by Gradient Descent](https://mlsanity.com/paper/2404.07049/) (2024)
- [A joint optimization approach to identifying sparse dynamics using least squares kernel collocation](https://mlsanity.com/paper/2511.18555/) (2025)

## Abstract (arXiv)

This paper presents a machine learning framework for Bayesian systems identification from noisy, sparse and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of recent developments in differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo. This allows us to efficiently infer posterior distributions over plausible models with quantified uncertainty, while the use of sparsity-promoting priors enables the discovery of interpretable and parsimonious representations for the underlying latent dynamics. A series of numerical studies is presented to demonstrate the effectiveness of the proposed methods including nonlinear oscillators, predator-prey systems, chaotic dynamics and systems biology. Taken all together, our findings put forth a novel, flexible and robust workflow for data-driven model discovery under uncertainty.