# Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification

arXiv: https://arxiv.org/abs/2305.08942 · HTML: https://mlsanity.com/paper/2305.08942/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2023.

## Built on by (most cited first, 4 of 4)

- [Fast data inversion for high-dimensional Ornstein-Uhlenbeck processes from noisy measurements](https://mlsanity.com/paper/2501.01324/) (2025)
- [Perfecting Liquid-State Theories with Machine Intelligence](https://mlsanity.com/paper/2311.05167/) (2023)
- [Unsupervised cell segmentation by fast Gaussian Processes](https://mlsanity.com/paper/2505.18902/) (2025)
- [Long-Term Probabilistic Forecast of Vegetation Conditions Using Climate Attributes in the Four Corners Region](https://mlsanity.com/paper/2601.16347/) (2026)

## Abstract (arXiv)

Data-driven modeling is useful for reconstructing nonlinear dynamical systems when the underlying process is unknown or too expensive to compute. Having reliable uncertainty assessment of the forecast enables tools to be deployed to predict new scenarios unobserved before. In this work, we first extend parallel partial Gaussian processes for predicting the vector-valued transition function that links the observations between the current and next time points, and quantify the uncertainty of predictions by posterior sampling. Second, we show the equivalence between the dynamic mode decomposition and the maximum likelihood estimator of the linear mapping matrix in the linear state space model. The connection provides a {probabilistic generative} model of dynamic mode decomposition and thus, uncertainty of predictions can be obtained. Furthermore, we draw close connections between different data-driven models for approximating nonlinear dynamics, through a unified view of generative models. We study two numerical examples, where the inputs of the dynamics are assumed to be known in the first example and the inputs are unknown in the second example. The examples indicate that uncertainty of forecast can be properly quantified, whereas model or input misspecification can degrade the accuracy of uncertainty quantification.