Bayesian Probabilistic Numerical Methods

Cockayne, Jon; Oates, Chris J.; Sullivan, Tim; Girolami, Mark

doi:10.1137/17m1139357

Cited by 115 publications

(127 citation statements)

References 109 publications

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“…Remark 3. Similar to our approach, Probabilistic Numerical Methods (PNMs) [63,64,65] take a statistical point of view of classical numerical methods (e.g. a finite element solver) that treat the output as a point estimate of the true solution.…”

Section: Probabilistic Surrogates With Reverse Kl Formulationmentioning

confidence: 99%

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Zhu

Zabaras

Koutsourelakis

et al. 2019

Journal of Computational Physics

761

458

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Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.

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Section: Probabilistic Surrogates With Reverse Kl Formulationmentioning

confidence: 99%

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Zhu

Zabaras

Koutsourelakis

et al. 2019

Journal of Computational Physics

761

458

View full text Add to dashboard Cite

show abstract

“…Albeit simple, this example aims to demonstrate the basic capabilities of the proposed methodology in propagating uncertainty through non-linear partial differential equations. In contrast to previous approaches to inferring solutions of partial differential equations from data [44,45,46,47], the proposed methodology does not rely on Gaussian assumptions, and it can directly tackle nonlinear problems without any need for linearization.…”

Section: A Pedagogical Examplementioning

confidence: 99%

Adversarial uncertainty quantification in physics-informed neural networks

Yang

Perdikaris

2019

Journal of Computational Physics

326

168

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We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physicsinformed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial inference procedure for training them on data, while constraining their predictions to satisfy given physical laws expressed by partial differential equations. Such physics-informed constraints provide a regularization mechanism for effectively training deep generative models as surrogates of physical systems in which the cost of data acquisition is high, and training data-sets are typically small. This provides a flexible framework for characterizing uncertainty in the outputs of physical systems due to randomness in their inputs or noise in their observations that entirely bypasses the need for repeatedly sampling expensive experiments or numerical simulators. We demonstrate the effectiveness of our approach through a series of examples involving uncertainty propagation in non-linear conservation laws, and the discovery of constitutive laws for flow through porous media directly from noisy data.

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“…Coalescence of some of the points would result in a quadrature rule that uses also evaluations of derivatives of the integrand. (14) where k (1) x is the kernel derivative defined in (11).…”

Section: Weights For Locally Optimal Pointsmentioning

confidence: 99%

On the positivity and magnitudes of Bayesian quadrature weights

2019

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This article reviews and studies the properties of Bayesian quadrature weights, which strongly affect stability and robustness of the quadrature rule. Specifically, we investigate conditions that are needed to guarantee that the weights are positive or to bound their magnitudes. First, it is shown that the weights are positive in the univariate case if the design points locally minimise the posterior integral variance and the covariance kernel is totally positive (e.g., Gaussian and Hardy kernels). This suggests that gradient-based optimisation of design points may be effective in constructing stable and robust Bayesian quadrature rules. Secondly, we show that magnitudes of the weights admit an upper bound in terms of the fill distance and separation radius if the RKHS of the kernel is a Sobolev space (e.g., Matérn kernels), suggesting that quasi-uniform points should be used. A number of numerical examples demonstrate that significant generalisations and improvements appear to be possible, manifesting the need for further research.

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Bayesian Probabilistic Numerical Methods

Cited by 115 publications

References 109 publications

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

Adversarial uncertainty quantification in physics-informed neural networks

On the positivity and magnitudes of Bayesian quadrature weights

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