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A probabilistic graphical model foundation for enabling predictive digital twins at scale

Nature Computational Science volume 1, pages 337–347 (2021)Cite this article

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A preprint version of the article is available at arXiv.

Abstract

A unifying mathematical formulation is needed to move from one-off digital twins built through custom implementations to robust digital twin implementations at scale. This work proposes a probabilistic graphical model as a formal mathematical representation of a digital twin and its associated physical asset. We create an abstraction of the asset–twin system as a set of coupled dynamical systems, evolving over time through their respective state spaces and interacting via observed data and control inputs. The formal definition of this coupled system as a probabilistic graphical model enables us to draw upon well-established theory and methods from Bayesian statistics, dynamical systems and control theory. The declarative and general nature of the proposed digital twin model make it rigorous yet flexible, enabling its application at scale in a diverse range of application areas. We demonstrate how the model is instantiated to enable a structural digital twin of an unmanned aerial vehicle (UAV). The digital twin is calibrated using experimental data from a physical UAV asset. Its use in dynamic decision-making is then illustrated in a synthetic example where the UAV undergoes an in-flight damage event and the digital twin is dynamically updated using sensor data. The graphical model foundation ensures that the digital twin calibration and updating process is principled, unified and able to scale to an entire fleet of digital twins.

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Fig. 1: Conceptual model of a physical asset and its associated digital twin.
Fig. 2: A mathematical abstraction of a digital twin and its associated physical asset.
Fig. 3: Probabilistic graphical model for the asset–twin system.
Fig. 4: Evolution of a UAV structural digital twin using the probabilistic graphical model formulation.
Fig. 5: Numerical results for the second calibration step (t = 2).
Fig. 6: Numerical results for the online phase of the simulated UAV mission.

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Data availability

Experimental data acquired during the calibration experiments of this study are available in the public repository UAV-Experimental-Calibration30. In all cases, the data generation and manipulation process is explained in detail in the Methods. Source data are provided with this paper.

Code availability

The code used to perform the calibration procedure is available in the public repository UAV-Experimental-Calibration30. This code, when combined with the provided experimental data, can be used to generate Fig. 5 and Extended Data Fig. 1. Additionally, code used to implement the in-flight health monitoring simulation is provided in the public repository UAV-Digital-Twin31. This simulation code was used to generate the data in Fig. 6. The structural analysis software used to generate the results in this paper is Akselos Integra v4.5.9 (https://akselos.com/). Because the Akselos Integra software is proprietary and was used under license, we are unable to provide its source code. Instead, the model output data are provided directly in the repositories.

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Acknowledgements

This work was supported in part by AFOSR grant no. FA9550-16-1-0108 under the Dynamic Data Driven Application Systems Program, the SUTD-MIT International Design Center, the AFOSR MURI on managing multiple information sources of multi-physics systems award nos. FA9550-15-1-0038 and FA9550-18-1-0023, US Department of Energy grant no. DE-SC0021239 and the AEOLUS Center under US Department of Energy Applied Mathematics MMICC award no. DE-SC0019303.

Author information

Authors and Affiliations

  1. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Michael G. Kapteyn

  2. The Jessara Group, Austin, TX, USA

    Jacob V. R. Pretorius

  3. Oden Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, USA

    Karen E. Willcox

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  1. Michael G. Kapteyn
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  2. Jacob V. R. Pretorius
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  3. Karen E. Willcox
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Contributions

M.G.K. and K.E.W. developed the probabilistic graphical model formulations. J.V.R.P. performed the calibration experiments and worked with M.G.K. to process the experimental data. M.G.K. implemented the in-flight health monitoring simulation. M.G.K. and K.E.W. wrote the manuscript, with contributions from J.V.R.P. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Karen E. Willcox.

Ethics declarations

Competing interests

Jessara Group sensors were used in the UAV experimental work described in this Article. Co-author J.V.R.P. is a co-founder of Jessara. Purchase of the sensors for use in the research was reviewed and approved in compliance with all applicable MIT policies and procedures.

Additional information

Peer review Information Nature Computational Science thanks Benjamin Herrmann, Rebecca Morrison, Omer San and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Jie Pan was the primary editor on this article and managed its editorial process and peer review in collaboration with the rest of the editorial team.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Calibration results for the UAV digital twin.

The first row shows prior information defining the initial estimate for each entry in the digital state. The second row shows posterior distributions for each entry in the digital state, which are the result of assimilating experimental data acquired via calibration experiments. We use \({\mathcal{N}}(\mu ,\sigma )\) to denote a Normal distribution with mean μ and standard deviation σ. For sample distributions we report the sample mean followed by the sample standard deviation in parentheses.

Source data

Source data

Source Data Fig. 5

Statistical source data.

Source Data Fig. 6

Statistical source data.

Source Data Extended Data Fig. 1

Statistical source data.

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Kapteyn, M.G., Pretorius, J.V.R. & Willcox, K.E. A probabilistic graphical model foundation for enabling predictive digital twins at scale. Nat Comput Sci 1, 337–347 (2021). https://doi.org/10.1038/s43588-021-00069-0

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