Use of Surrogate Models for Continuous Optimal Experimental Design

Abstract

Models of physical systems typically include parameters that require estimation from noisy experimental observations. One such example of this comes from the field of biomechanical modelling, where there is a need to estimate the mechanical properties of soft tissues, such as their stiffnesses, by observing their deformation under a known mechanical load. Such problems provide the primary motivation for the work in this thesis. These are challenging as there are innumerably many loads one can apply to soft tissues to infer their stiffness, but it is not immediately obvious which load is ‘best’. Moreover, in a clinical context, it is preferable to identify these stiffness parameters using as few experiments as possible, as measurements are typically costly and time-consuming. The statistical field of Optimal Experimental Design (OED) provides a framework for deciding which particular experiment to carry out to best estimate a target parameter. Although many different OED methodologies exist, many of these approaches are practically limited either by their overly-restrictive assumptions, their computational cost, or their requirement for iterative sets of experiments. Here we develop a flexible OED workflow that can be applied to a wide range of physical systems. More specifically, this workflow leverages Gaussian Process (GP) surrogate models, amortised variational inference, and local linearisation approximations to the posterior to efficiently minimise the so-called Average Posterior Entropy (APE) through stochastic gradient descent. Using numerical experiments, we then apply this workflow to the simple prototype biomechanics problem of how one ought to orientate a soft cantilever beam to infer its stiffness by observing its deformation under gravity. Applying our framework to this problem, we show that the ‘naive’ design of orientating the beam horizontally is not the optimal experiment to identify a neo-Hookean cantilever beam’s stiffness. Instead, the angle at which the beam ought to be orientated lies somewhere between 110◦ to 130◦ above the direction of gravity, with the exact optimal angle being determined by how soft the experimenter believes the beam is prior to performing any experiments. Even though the scenario considered in this work is relatively simple, the workflow itself should be able to be extended to more complex problems in higher-dimensional parameter and design spaces, particularly biomechanics models.