Sparsifying priors for Bayesian uncertainty quantification in model discovery

@article{Hirsh2021SparsifyingPF,
  title={Sparsifying priors for Bayesian uncertainty quantification in model discovery},
  author={Seth M. Hirsh and David A. Barajas-Solano and J. Nathan Kutz},
  journal={Royal Society Open Science},
  year={2021},
  volume={9}
}
We propose a probabilistic model discovery method for identifying ordinary differential equations governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, where models are expressed as sparse linear combinations of pre-specified candidate functions. Promoting parsimony through sparsity leads to interpretable models that generalize to unknown data. Instead of targeting point estimates of the SINDy coefficients… 
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References

SHOWING 1-10 OF 77 REFERENCES

Bayesian differential programming for robust systems identification under uncertainty

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…

Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

A variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al is developed, which can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

Bayesian system ID: optimal management of parameter, model, and measurement uncertainty

A first principles derivation of appropriate objective formulations for system identification based on probabilistic principles is studied and the resulting inference objective is compared to those used by emerging data-driven methods based on dynamic mode decomposition (DMD) and system identification of nonlinear dynamics (SINDy).

Dirichlet–Laplace Priors for Optimal Shrinkage

This article proposes a new class of Dirichlet–Laplace priors, which possess optimal posterior concentration and lead to efficient posterior computation.

The Bayesian Lasso

The Lasso estimate for linear regression parameters can be interpreted as a Bayesian posterior mode estimate when the regression parameters have independent Laplace (i.e., double-exponential) priors.…

BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS

Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector.

Sparsity information and regularization in the horseshoe and other shrinkage priors

A concept of effective number of nonzero parameters is introduced, an intuitive way of formulating the prior for the global hyperparameter based on the sparsity assumptions is shown, and the previous default choices are argued to be dubious based on their tendency to favor solutions with more unshrunk parameters than the authors typically expect a priori.

POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES

This paper considers Bayesian counterparts of the classical tests for good- ness of fit and their use in judging the fit of a single Bayesian model to the observed data. We focus on posterior…

Inferring Biological Networks by Sparse Identification of Nonlinear Dynamics

This method, implicit-SINDy, succeeds in inferring three canonical biological models: 1) Michaelis-Menten enzyme kinetics; 2) the regulatory network for competence in bacteria; and 3) the metabolic network for yeast glycolysis.

A Unified Sparse Optimization Framework to Learn Parsimonious Physics-Informed Models From Data

A flexible ML-based framework for learning governing models for physical systems from data that addresses three open challenges in scientific problems and data sets, including robust handling of outliers and corrupt data within noisy sensor measurements, parametric dependencies in candidate library functions, and the imposition of physical constraints.
...