Background to Challenge. This challenge is based on a coupled multiphysics problem that is currently implemented within COMSOL. The model represents heat and mass transfer and solid mechanics processes during the baking of a dough product, and has around 13 input parameters describing the underlying physical processes. The output consists of three time series: (i) temperature and (ii) fraction of liquid converted to vapour (iii) deformation of the shape. Sensitivity analysis indicates at least 5 of the model inputs to be the most important, namely the rate of evaporation from liquid to vapour, the intrinsic permeability of the dough, the glass transition temperature, rheological constants and the vapour porosity. These parameters are hard to determine experimentally, so must be calibrated using available experimental data for outputs (i) (ii) and (iii) at a number of specific times.

Expected Outcomes of Study Group. PepsiCo would like to understand whether there is a unique choice of the 5 key parameters that corresponds to given model outputs, and in general how to better calibrate their model on noisy data.

Proposed Approach. The non-uniqueness problem can be posed as a penalised optimisation problem: seek another solution that is not close to the solution(s) already known. We note that the non-linearity of the model is what makes this interesting; if the input-output map were linear this would be a trivial problem. It would be easiest to prototype the method on a cheap surrogate model but design the method with an API that could also interface with the full, expensive model to allow validation after the Study Group has finished. Bayesian optimisation approaches might be flexible enough to naturally take in the cost-accuracy tradeoff, and a library of past evaluations, while also handling a transition from wide exploration of the parameter space to exploiting candidate optima. The API should also be flexible enough to handle either the time series formulation or the terminal state version of the problem.

Possible Extensions. We could ask the Study Group to reconsider the choice of the 5 most important input parameters, i.e. re-run the sensitivity analysis either with the same methodology as a validation exercise, or with a different choice of sensitivity indices, or even seek a low-dimensional active subspace for the system, not necessarily aligned to the coordinate axes.