Bayesian differential programming for robust systems identification under uncertainty

Yang, Yibo; Bhouri, Mohamed Aziz; Perdikaris, Paris

doi:10.48550/arxiv.2004.06843

Cited by 2 publications

(4 citation statements)

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“…We note that the range of predicted model states is much narrower than for the Lotka-Volterra model, which is expected due to the lower noise level present in these measurements. We also emphasize that these PPDs are much tighter than those presented in [66] for this test case, even though we train rh-SINDy and ss-SINDy with substantially less data than in that work. Furthermore, it can be seen that the test data lies within the 90% credibility intervals of the PPDs of each state.…”

Section: Nonlinear Oscillator and Model Indeterminacymentioning

confidence: 85%

“…Discovery of governing equations plays a fundamental role in the development of physical theories. With increasing computing power and data availability in recent years, there have been substantial efforts to identify the governing equations directly from data [7,51,66]. There has been particular emphasis on parsimonious representations because they have the benefits of promoting interpretibility and generalizing well to unknown data [2,9,8,38,43,46,60,63].…”

Section: Introductionmentioning

confidence: 99%

“…Bayesian methods have been widely used for uncertainty quantification in time series models, with applications to weather forecasting [1,20,67], disease modeling [3,35,68], traffic flow [13,55,70], and finance [24,58,65], among many others. More recently, these methods have been incorporated into model discovery frameworks, exhibiting state-of-the-art performance for system identification in the presence of noise [21,42,66]. Although these methods provide a range of possible values, realizations of these models are in general not sparse and consequently lack the capability to identify relevant terms in the model.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery

Hirsh¹,

Barajas-Solano²,

Kutz³

2021

Preprint

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We propose a probabilistic model discovery method for identifying ordinary differential equations (ODEs) governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, in which target ODE models are expressed as a sparse linear combinations of pre-specified candidate functions. Promoting parsimony through sparsity in SINDy leads to interpretable models that generalize to unknown data. Instead of targeting point estimates of the SINDy (linear combination) coefficients, in this work we estimate these coefficients via sparse Bayesian inference. The resulting method, uncertainty quantification SINDy (UQ-SINDy), quantifies not only the uncertainty in the values of the SINDy coefficients due to observation errors and limited data, but also the probability of inclusion of each candidate function in the linear combination. UQ-SINDy promotes robustness against observation noise and limited data, interpretability (in terms of model selection and inclusion probabilities), and generalization capacity for out-of-sample forecast. Sparse inference for UQ-SINDy employs Markov Chain Monte Carlo, and we explore two sparsifying priors: the spike-and-slab prior, and the regularized horseshoe prior. We apply UQ-SINDy to synthetic nonlinear data sets from a Lotka-Volterra model and a nonlinear oscillator, and to a real-world data set of lynx and hare populations. We find that UQ-SINDy is able to discover accurate and meaningful models even in the presence of noise and limited data samples.

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Section: Nonlinear Oscillator and Model Indeterminacymentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery

Hirsh¹,

Barajas-Solano²,

Kutz³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…These include symbolic regression [6,29,24], sequential thresholding [8,27,28], information theoretic methods [2,15], relaxation methods, [31,45], and constrained sparse optimization [34]. Bayesian methods for nonlinear system identification [39,21] including ARD have been applied to the nonlinear system identification problem for improved robustness in the case of low data [40] and for uncertainty quantification [42,43,12]. The critical challenge for any library method for nonlinear system identification is learning the correct set of active terms.…”

Section: Introductionmentioning

confidence: 99%

Sparse Methods for Automatic Relevance Determination

Rudy,

Sapsis

2020

Preprint

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This work considers methods for imposing sparsity in Bayesian regression with applications in nonlinear system identification. We first review automatic relevance determination (ARD) and analytically demonstrate the need to additional regularization or thresholding to achieve sparse models. We then discuss two classes of methods, regularization based and thresholding based, which build on ARD to learn parsimonious solutions to linear problems. In the case of orthogonal covariates, we analytically demonstrate favorable performance with regards to learning a small set of active terms in a linear system with a sparse solution. Several example problems are presented to compare the set of proposed methods in terms of advantages and limitations to ARD in bases with hundreds of elements. The aim of this paper is to analyze and understand the assumptions that lead to several algorithms and to provide theoretical and empirical results so that the reader may gain insight and make more informed choices regarding sparse Bayesian regression.

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Bayesian differential programming for robust systems identification under uncertainty

Cited by 2 publications

References 0 publications

Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery

Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery

Sparse Methods for Automatic Relevance Determination

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