A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables

Rixner, Maximilian; Koutsourelakis, Phaedon‐Stelios

doi:10.1016/j.jcp.2021.110218

Cited by 11 publications

(2 citation statements)

References 45 publications

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“…The "noise" parameter σ c can be used to account for the intensity of the enforcement of the virtual observations and represents the tolerance parameter with which the constraints would be enforced in a deterministic setting. We note that the concept of virtual observables is not restricted to physical constraints but could also be applied to residuals of temporal discretization schemes [6] or of PDEs [16]. In both of this cases, it is shown that the incorporation of virtual observables can reduce the amount of training data required and enable training in the Small Data regime.…”

Section: Virtual Observablesmentioning

confidence: 99%

Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime

Kaltenbach¹,

Koutsourelakis²

2021

Preprint

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The data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a probabilistic perspective that simultaneously identifies predictive, lower-dimensional coarse-grained (CG) variables as well as their dynamics. We make use of the expressive ability of deep neural networks in order to represent the right-hand side of the CG evolution law. Furthermore, we demonstrate how domain knowledge that is very often available in the form of physical constraints (e.g. conservation laws) can be incorporated with the novel concept of virtual observables. Such constraints, apart from leading to physically realistic predictions, can significantly reduce the requisite amount of training data which enables reducing the amount of required, computationally expensive multiscale simulations (Small Data regime). The proposed state-space model is trained using probabilistic inference tools and, in contrast to several other techniques, does not require the prescription of a fine-to-coarse (restriction) projection nor time-derivatives of the state variables. The formulation adopted is capable of quantifying the predictive uncertainty as well as of reconstructing the evolution of the full, fine-scale system which allows to select the quantities of interest a posteriori. We demonstrate the efficacy of the proposed framework in a high-dimensional system of moving particles.

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Section: Virtual Observablesmentioning

confidence: 99%

Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime

Kaltenbach¹,

Koutsourelakis²

2021

Preprint

Self Cite

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“…These models integrate prior knowledge of the underlying physics, governing equations, or constraints into the modeling process. Due to the additional information also higher dimensional inverse problems or uncertainty quantification tasks can be tackled as demonstrated in [29][30][31]. However, for the application of these approaches, the surrogate model has to be adjusted to each investigated forward model, which is often inconvenient when, like in our case, quite different classes of problems should be tackled.…”

Section: Introductionmentioning

confidence: 99%

Solving Bayesian inverse problems with expensive likelihoods using constrained Gaussian processes and active learning

Dinkel,

Geitner,

Robalo Rei

et al. 2024

Inverse Problems

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Solving inverse problems using Bayesian methods can become prohibitively expensive when likelihood evaluations involve complex and large scale numerical models. A common approach to circumvent this issue is to approximate the forward model or the likelihood function with a surrogate model. But also there, due to limited computational resources, only a few training points are available in many practically relevant cases. Thus, it can be advantageous to model the additional uncertainties of the surrogate in order to incorporate the epistemic uncertainty due to limited data. In this paper, we develop a novel approach to approximate the log likelihood by a constrained Gaussian process based on prior knowledge about its boundedness. This improves the accuracy of the surrogate approximation without increasing the number of training samples. Additionally, we introduce a formulation to integrate the epistemic uncertainty due to limited training points into the posterior density approximation. This is combined with a state of the art active learning strategy for selecting training points, which allows to approximate posterior densities in higher dimensions very efficiently. We demonstrate the fast convergence of our approach for a benchmark problem and infer a random field that is discretized by 30 parameters using only about 1000 model evaluations. In a practically relevant example, the parameters of a reduced lung model are calibrated based on flow observations over time and voltage measurements from a coupled electrical impedance tomography simulation.

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Manifold learning for coarse-graining atomistic simulations: Application to amorphous solids

et al. 2021

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A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables

Cited by 11 publications

References 45 publications

Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime

Physics-aware, deep probabilistic modeling of multiscale dynamics in the Small Data regime

Solving Bayesian inverse problems with expensive likelihoods using constrained Gaussian processes and active learning

Manifold learning for coarse-graining atomistic simulations: Application to amorphous solids

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