# Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inference

arXiv: https://arxiv.org/abs/2401.16943 · HTML: https://mlsanity.com/paper/2401.16943/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 77 references that are in the map.
- First posted on arXiv 30 Jan 2024.

## Builds on (most cited first, 9 of 9)

- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [A flexible state space model for learning nonlinear dynamical systems](https://mlsanity.com/paper/1603.05486/) (2016)
- [Joint reconstruction and segmentation of noisy velocity images as an inverse Navier-Stokes problem](https://mlsanity.com/paper/2207.01368/) (2022)
- [Bayesian data-driven discovery of partial differential equations with variable coefficients](https://mlsanity.com/paper/2102.01432/) (2021)

## Built on by (most cited first, 9 of 9)

- [Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy data](https://mlsanity.com/paper/2402.15357/) (2024)
- [VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification](https://mlsanity.com/paper/2405.20905/) (2024)
- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Physics-Informed Neural Networks with Unknown Partial Differential Equations: an Application in Multivariate Time Series](https://mlsanity.com/paper/2503.20144/) (2025)
- [Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identification](https://mlsanity.com/paper/2604.20141/) (2026)
- [Model Based and Physics Informed Deep Learning Neural Network Structures](https://mlsanity.com/paper/2408.07104/) (2024)
- [Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics](https://mlsanity.com/paper/2603.05201/) (2026)
- [One-shot learning for the complex dynamical behaviors of weakly nonlinear forced oscillators](https://mlsanity.com/paper/2604.15181/) (2026)

## Abstract (arXiv)

This study presents a Bayesian maximum textita~posteriori (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint maximum textita~posteriori (JMAP) and variational Bayesian approximation (VBA), are compared to the LASSO, ridge regression and SINDy algorithms for sparse regression, by application to several dynamical systems with added Gaussian or Laplace noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or ``Gaussian norm'' ||vecy-hatvecy||2_M-1 = (vecy-hatvecy)^top M-1 (vecy-hatvecy), where vecy is a vector variable, hatvecy is its estimator and M is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection for the different systems and noise models examined.