34 On Digital Twins, Mirrors and Virtualisations 295 approach model bias, or epistemic uncertainty in general. For this reason, some of the definitions given here are independent of whatever uncertainty theory ultimately dominates in a given context. As long as an uncertainty theory singles out some most highly indicatedmodel from the population of possible choices, one can base the analysis on the -mirror for that single model. For example, in a Bayesian framework, one can apply the idea to theMaximum a Posteriori (MAP) model. Of course, any theorems in the general theory will have to be proved independently for each uncertainty specification. In many ways, the paper presents a wish list; however, it does so in the real hope that the wishes can come true—that the required theory can come together. The paper presents only the sketchiest arguments as to how the various ‘theorems’ might be proved, or how the relevant estimates could be made; this is because the current authors do not have anything like the complete range of abilities/skills that will be needed in order to assemble the theory. In many ways the paper is intended as a rallying call to the academic community; the skills needed will come from a range of disciplines: pure and applied mathematics, physics, computer science (particularly machine learning) and engineering. The authors here believe that a framework can come together which is more than the sum of its parts and that can be of lasting value in the pursuit of effective computer models, and particularly in the construction of digital twins. Acknowledgement The authors would like to acknowledge the support of the UK Engineering and Physical Sciences Research Council (EPSRC) through grant reference numbers EP/J016942/1 and EP/K003836/2. References 1. Tuegel, E.J., Ingraffea, A.R., Eason, T.G., Spottswood, S.M.: Reengineering aircraft structural life prediction using a digital twin. Int. J. Aerosp. Eng. 2011, 154798 (2011) 2. Datta, S.P.A.: Emergence of digital twins - is this the march of reason. J. Innov. Manag. 5, 14–33 (2017) 3. Grieves, M., Vickers, J.: Digital-Twin: Mitigating Unpredictable, Undesirable Emergent Behavior in Complex Systems, pp. 85–113. Springer, Switzerland (2017) 4. Wagg, D.J., Gardner, P., Barthorpe, R.J., Worden, K.: On key technologies for realising digital-twins for structural dynamics applications. In: Proceedings of the 37th IMAC, the International Modal Analysis Conference, Orlando (2019) 5. Li, C., Mahadevan, S.: Role of calibration, validation and relevance in multi-level uncertainty integration. Reliab. Eng. Syst. Saf. 148, 32–43 (2016) 6. Nagel, J., Sudret, B.: A unified framework for multilevel uncertainty quantification in Bayesian inverse problems. Probab. Eng. Mech. 43, 68–84 (2015) 7. Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. 63, 425–464 (2001) 8. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2007) 9. Cherkassky, V., Mulier, F.M.: Learning from Data: Concepts, Theory and Methods. Wiley, Hoboken (1998) 10. Worden, K., Barthorpe, R.J., Cross, E.J., Dervilis, N., Holmes, G.R., Manson, G., Rogers, T.J.: On evolutionary system identification with applications to nonlinear benchmarks. Mech. Syst. Signal Process. 112, 194–232 (2018) 11. Worden, K., Hensman, J.J.: Parameter estimation and model selection for a class of hysteretic systems using Bayesian inference. Mech. Syst. Signal Process. 32, 153–169 (2012) 12. Abdessalem, A.B., Dervilis, N., Wagg, D.J., Worden, K.: Model selection and parameter estimation in structural dynamics using approximate Bayesian computation. Mech. Syst. Signal Process. 32, 306–325 (2018) 13. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT, Cambridge (2006) 14. Worden, K.: Some thoughts on model validation for nonlinear systems. In: Proceedings of 3rd International Conference on Identification in Engineering Systems, Swansea, pp. 142–154 (2002) 15. Turing, A.: Computing machinery and intelligence. Mind 236, 433–460 (1950) 16. Ladeveze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20, 485–509 (1983) 17. Ainsworth, M., Tinsley, J.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Eng. 142, 1–88 (1997) 18. Gawthrop, P., Neild, S., Wallace, M., Wagg, D.J.: Robust real-time substructuring techniques for lightly-damped systems. Struct. Control. Health Monit. 14, 591–600 (2007) 19. Farrar, C.R., Worden, K.: Structural Health Monitoring: A Machine Learning Perspective. Wiley, Hoboken (2012) 20. Cappell, S., Ranicki, A., Rosenberg, J.: Surveys on Surgery Theory: Volume 1. Princeton University, Princeton (2000)
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