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Uncertainty quantification in digital twin engineering

Project Description

Computer model predictions of prototypes are only as good as their underlying assumptions. Design decisions need to account for manufacturing and environmental uncertainties in a robust yet efficient manner. Robust to uncertainties safe-guards designs against unlikely but possible scenarios, yet an overly conservative design undermines efficiency. A balance between robustness and efficiency is essential, which requires uncertainties and their effects to be properly quantified so that risk-informed and defensible design decisions can be made.

This Ph.D. project is funded in support of a multi-disciplinary EPSRC program grant for digital twins for improved dynamic design. Key questions fundamental in establishing a practical digital twin framework include:

• How do we decide which engineering and scientific models to use?

• How do we quantify the uncertainties in the models’ inputs, and the uncertainties in the form of the models themselves?

• How should models be validated against empirical observations?

• How can we interpret the model output to make an engineering decision?

These questions will be addressed through comprehensive uncertainty quantification that respects the distinction between two main kinds of uncertainty: aleatory uncertainty is stochasticity or variability across space, time or components; and epistemic uncertainty is imprecision or incertitude about conditions, parameters, or model structure that arises from incomplete knowledge. The project will combine sampling-based strategies such as Monte Carlo simulation and constraint analysis such as interval bounding and projection.

Monte Carlo simulation is especially useful for characterising aleatory uncertainties. Its approximation error is determined by the number of samples, not by the number of uncertain variables. It can be applied to black-box models, which means it can be applied without “intrusively” interrogating the details of the calculations within the model.

Constraint analysis and related methods in imprecise probabilities can be used to characterise and propagate epistemic uncertainties. However, these methods are generally intrusive in the sense that they must be applied to each binary mathematical operation (+, -, etc) within a simulation, causing difficulties with black-box models.

The project will develop strategies that work efficiently and scalably with aleatory, epistemic, and mixed uncertainties. More information about the project consortium can be found at

Requirements and application
The successful candidate will have a master’s degree in engineering, mathematics, theoretical physics, or physics. Programming experience is welcome.

Institute for Risk and Uncertainty (
The Liverpool Institute for Risk & Uncertainty (Risk Institute) was founded in 2011 at the University of Liverpool as a multi-disciplinary, large-scale Research Institute. It combines expertise in quantifying, managing and mitigating risk and uncertainty from over 10 disciplines across the University including architecture, engineering, environmental sciences, the institute for infection and global health, financial and actuarial mathematics, computer science, electrical engineering, economics and finance, management, social sciences, psychology and law. The focus is on the comprehensive understanding of uncertainties and associated risks as key issues in the performance assessment of complex systems, and in the development of proper mitigation strategies. With strong ties to industry, applications are pursued in a wide range of areas such as building design, climate change analysis, reliability engineering, software reliability, material science, financial modelling, socio-political harm reduction and critical incident management.

Funding Notes

The PhD Studentship (Tuition fees + stipend of £ 14,553 annually over 3 years) is available.
Candidates from EU countries other than the UK are eligible for awards restricted to the payment of tuition fees only.
UK/EU candidates must be ordinarily resident in the UK/EU. Candidates who are resident in the UK, will not be eligible on residence ground.

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