1. Introduction
Differential equation models are central to modern science and engineering.
They are used to simulate blood flow in cardiovascular systems, predict disease progression, represent ocean and climate processes, and evaluate interventions before they are tested in the real world. However, these models can be computationally expensive, especially when they must be repeatedly evaluated for parameter inference, uncertainty quantification, model calibration, sensitivity analysis, or decision support. This creates a sustainability challenge: how can we use machine learning, AI and statistical learning to reduce computational cost and energy use while maintaining accurate and reliable predictions, with calibrated uncertainty?
This PhD project will develop sustainable emulation-assisted Bayesian inference methods for differential equation models, with applications in healthcare and climate modelling. We will focus on statistical emulators that provide fast and reliable approximations to expensive simulators while quantifying uncertainty in their predictions. The central aim is to understand how different emulation strategies can reduce computational cost without producing biased parameter estimates and mis-calibrated uncertainty.
Research questions that the project will focus on are as follows:
Resource allocation: how should we allocate expensive simulator evaluations when constructing and refining cheap emulators for downstream tasks? This includes comparing emulators of simulator outputs and emulators of derived objective functions to understand which gives the best trade-off between cost, accuracy and calibrated uncertainty.
Multi-fidelity modelling: how can we combine cheap approximate simulators/emulators with expensive higher-fidelity simulators/emulators to reduce the number of costly model runs within a parameter inference scheme? This may involve formulating the inverse problem within the classical autoregressive Gaussian-process co-kriging framework, modern deep or nonlinear multi-fidelity Gaussian processes, and active learning strategies that decide whether and when high-fidelity or low-fidelity simulations are most valuable.
Incorporating physics: how can speed and accuracy be improved by replacing data-driven emulators by physics-informed emulators? Here, the differential-equation structure is estimated alongside parameters, thus bypassing costly numerical evaluations of the differential equations altogether.
Foundations and Methodology
A major emphasis will be on Bayesian uncertainty quantification. In many scientific applications, a fast but inaccurate emulator is dangerous: it can lead to incorrect parameter estimates, unreliable predictions, and potentially poor decisions. The project will therefore study approaches capable of integrating multiple sources of uncertainty, such as observation noise, emulator uncertainty, numerical solver error, model discrepancy, low-/high-fidelity mismatch, and uncertainty in physical parameters. Calibration diagnostics, posterior predictive checking, simulation-based calibration, and decision-relevant uncertainty metrics will be used to assess whether the resulting uncertainty quantification is trustworthy.
The methodological work will be motivated by two application areas. In healthcare, cardiovascular models, such as 0D Windkessel models and 1D haemodynamic models, provide a natural test bed for inferring parameters such as vessel wall stiffness or boundary condition parameters from sparse pressure, flow or imaging data. In climate and environmental modelling, ocean-column or climate-component models provide a complementary test bed, where the aim may be to infer unknown parameters such as vertical mixing parameters, uncertain fluxes, or biogeochemical rates from sparse observations or expensive simulation output.
The student will gain expertise in Gaussian processes, statistical emulation, Bayesian inverse problems, multi-fidelity modelling, physics-informed machine learning, and scientific computing.
References:
Williams, C. K. I. and Rasmussen, C. E. (2006). Gaussian Processes for Machine Learning. MIT Press.
Kennedy, M. C. and O’Hagan, A. (2001). Bayesian calibration of computer models. Journal of the Royal Statistical Society: Series B, 63(3), 425–464.
Kennedy, M. C. and O’Hagan, A. (2000). Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1), 1–13.
Perdikaris, P., Raissi, M., Damianou, A., Lawrence, N. D. and Karniadakis, G. E. (2017). Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling. Proceedings of the Royal Society A, 473(2198), 20160751.
Raissi, M. and Karniadakis, G. E. (2017). Machine learning of linear differential equations using Gaussian processes. Journal of Computational Physics, 348, 683–693.
Li, Z., Yang, S. and Wu, C. F. J. (2024). Parameter inference based on Gaussian processes informed by nonlinear partial differential equations. SIAM/ASA Journal on Uncertainty Quantification, 12(3), 964–1004.