LOCAL APPROXIMATION TO COMPLEX MODELS FOR EFFICIENT OPTIMISATION: APPLICATION TO CRYSTALLISATION PROCESSES
Systematic methods and tools for managing the complexity
Process Simulation & Optimization - III (T4-9c)
Keywords: mathematical modelling, local approximation, statistical modelling, Gaussian process model, simulation-based optimisation
Recent advances in mathematical modelling have enabled the development of detailed mechanistic models to support the design and analysis of chemical processes or products. However in those cases where the resulting model is highly complex, it can be extremely time consuming to solve the model numerically, thereby limiting its practical applicability. This is especially the case where such a complex model is used for the numerical optimisation of a process thereby necessitating the execution of a large number of evaluations of the model. To make such an optimisation task computationally tractable, it is very important to apply appropriate model simplification treatments within the optimisation framework.
In this research, the idea of approximating complex mechanistic models with statistical models in the context of optimisation is investigated. More specifically a number of methods previously developed in the field of simulation-based optimisation are considered. In particular, in this research, an iterative optimisation procedure is implemented, which makes use of a Gaussian process model (GPM) to locally approximate a complex mechanistic model (CMM). Within each iteration of the optimisation procedure, the GPM is trained using data generated from a number of executions of the CMM, where the inputs to the CMM are selected from a sub-region of the entire optimisation domain. It is thus assumed that the GPM is valid within this sub-region, where the optimisation is subsequently performed using the GPM instead of the CMM. The GPM computes both the mean and the standard deviation of its prediction, with the latter metric providing an indication as to the accuracy of the GPM. This information is utilised to determine the next sub-region for approximate modelling and for optimisation in a way that the improvement of the objective function and the improvement of the GPM’s accuracy are well balanced. This process continues until a particular stopping criterion is met.
A consequence of the use of the local approximation model is that the optimisation procedure requires fewer evaluations of the original CMM compared with the number required when the analysis is based solely on the CMM, hence making the optimisation computationally more efficient. This method treats the CMM as a black box when the CMM is being approximated. Therefore, unlike the model reduction – based methods, it virtually requires no knowledge about the structure of the CMM. This method also differs from those based on the global approximation of the CMM: it successively constructs local approximating models which are much easier to build in comparison to global models, especially when the dimension of the input space is high.
This procedure has successfully been applied to the optimisation of a batch crystallization process. The process is mechanistically modelled through a set of two-dimensional partial differential-algebraic equations. Its operation includes discontinuities, which significantly slow down the numerical integration, thereby impacting on the solution of the model. Consequently, the evaluation of this model is computationally very costly and the optimisation based on this model requires at least a few days to complete. By applying the local approximation based optimisation procedure, the required number of evaluations of the complex model is dramatically reduced, and the optimisation can be completed successfully within a few hours.
See the full pdf manuscript of the abstract.
Presented Wednesday 19, 12:00 to 12:20, in session Process Simulation & Optimization - III (T4-9c).
