A joint model-based design of experiments approach for the identification of Gaussian Process models in geological exploration

When searching for potential mining sites, accurately modelling mineral concentrations or rock qualities in the subsurface is a crucial task. However, drilling in these locations is an expensive process, so reliable interpolation and efficient sampling techniques are required (Rossi & Deutsch, 2014). Gaussian Processes (GPs), also known as Kriging models, were first developed in the mining industry in the 1950s and continue to be widely used in resource modelling (Sahimi, 2011). As the true nature of the subsurface is unknown, assumptions must be made about the kernel function, which describes correlation structures between probable distributions of spatial phenomena, and its parameters must be estimated. This is typically accomplished through expert judgement and exploratory data analysis of preliminary samples. Model predictions are updated iteratively as more drilling data becomes available, with a focus on balancing expected exploitation (high grade intercepts) and exploration (minimising the Kriging variance) (Jafrasteh & Suarez, 2020). However, problems can arise if the chosen kernel is incorrect or if high uncertainty affects parameters. This poster showcases a joint model-based design approach (Galvanin et al., 2016) aiming to optimise three objectives: 1) reducing parametric uncertainty; 2) increasing the exploration of the design space to avoid local optima; 3) maximising the distinguishability of candidate model predictions to identify the most suitable kernel function with the minimum number of samples. Two different kernels in an Ordinary Kriging GP were used as candidate models and in-silico data was generated using one kernel. Starting from some initial samples, the optimal design strategy iteratively determined sampling locations to maximise the distinguishability between model predictions with a constraint ensuring that each iteration reduces prediction variance. The correct model could be distinguished and the data approximated well with a limited number of drilling experiments while satisfactorily estimating kernel parameters

Mechanistic modelling of separating dispersions in pipes using model-based design of experiments techniques

This work presents a parametric study on a mechanistic model for separating liquid–liquid dispersions in pipes. The model considers drop-settling, drop-interface coalescence and drop-drop coalescence, predicting the evolution of four characteristic layers during separation. Parameter estimation, parametric sensitivity analysis (PSA), and model-based design of experiments (MBDoE) techniques are employed to acquire precise parameter estimates and propose optimal experimental conditions, thereby enhancing the accuracy of existing models. Experimental data from literature using oil-in-water dispersions are used for parameter estimation. PSA reveals regions of high sensitivity of the model outputs to uncertain parameters, which are corresponding to favourable sampling locations. Manipulating the mixture velocity, the dispersed phase fraction, and the layer heights at the inlet influences these sensitive regions. Clustered measurements around highly sensitive regions in the pipe enhance the information content they provide. MBDoE demonstrates that either of the A-, D-, or E-optimal experimental design criteria improves the expected parameter precision

An evolutionary approach to kinetic modelling inspired by Lamarckian inheritance

The mechanistic description of kinetic phenomena requires the construction of systems of differential and algebraic equations where a high number of parameters and state variables may be involved. The complexity associated with kinetic phenomena frequently leads to the construction of kinetic models characterised by some degree of approximation. Whenever an approximated model is falsified by observations, its mathematical structure should be evolved embracing the available experimental evidence [1]. Nonetheless, improving a model is a time and resource intensive task that heavily relies on the presence of experienced researchers. An evolutionary approach to kinetic modelling is proposed in this work which is inspired by the theory of evolution proposed by Jean-Baptiste Lamarck, namely the theory of Lamarckian inheritance [2]. The approach is illustrated qualitatively in the sketch in Figure 1. Lamarck states that the evolution of living beings is directly driven by their interaction with the environment. The use/disuse of an organ determines the evolution of that organ towards higher/lower complexity. The long neck of the giraffe is frequently reported as an example to explain Lamarck’s theory. Primitive giraffes with a short neck would strive to reach the highest leaves. This behaviour, driven by necessities of adaptation, would lead to an elongation of the neck during the giraffe’s lifetime and the characteristic of the long neck is inherited by the offspring. Lamarck’s theory is now widely dismissed. Nonetheless, the field of epigenetics stemmed directly from Lamarck’s philosophy and aims at explaining the complex mechanisms behind the hereditability of environment-driven phenotype changes [3]. In this work, the principles of Lamarckian inheritance are translated into a framework for kinetic model building. In the proposed framework, the evolution of model structures is data-driven. When the model is over-fitting, model parameters that are irrelevant for representing the data are removed from the model structure. When the model is under-fitting, relevant model parameters are evolved into more complex state-dependent expressions. A statistical index, namely a Model Modification Index (MMI), based on the Lagrange multipliers statistic [4], is proposed as a measure of model misspecification to support the evolution of approximated kinetic models towards higher levels of complexity. The use of the MMI is demonstrated in a simulated case study to diagnose misspecification in an approximated kinetic model of baker’s yeast growth [5]. Please click Additional Files below to see the full abstract

An exploratory model-based design of experiments approach to aid parameters identification and reduce model prediction uncertainty

The management of trade-off between experimental design space exploration and information maximization is still an open question in the field of optimal experimental design. In classical optimal experimental design methods, the uncertainty of model prediction throughout the design space is not always assessed after parameter identification and parameters precision maximization do not guarantee that the model prediction variance is minimized in the whole domain of model utilization. To tackle these issues, we propose a novel model-based design of experiments (MBDoE) method that enhances space exploration and reduces model prediction uncertainty by using a mapping of model prediction variance (G-optimality mapping). This explorative MBDoE (eMBDoE) named G-map eMBDoE is tested on two models of increasing complexity and compared against conventional factorial design of experiments, Latin Hypercube (LH) sampling and MBDoE methods. The results show that G-map eMBDoE is more efficient in exploring the experimental design space when compared to a standard MBDoE and outperforms classical design of experiments methods in terms of model prediction uncertainty reduction and parameters precision maximization

Autonomous kinetic model identification using optimal experimental design and retrospective data analysis: methane complete oxidation as a case study

Automation and feedback optimization are combined in a smart laboratory platform for the purpose of identifying appropriate kinetic models online. In the platform, model-based design of experiments methods are employed in the feedback optimization loop to design optimal experiments that generate data needed for rapid validation of kinetic models. The online sequential decision-making in the platform, involving selection of the most appropriate kinetic model structure followed by the precise estimation of its parameters is done by autonomously switching the respective objective functions to discriminate between competing models and to minimise the parametric uncertainty of an appropriate model. The platform is also equipped with data analysis methods to study the behaviour of models within their uncertainty limits. This means that the platform not only facilitates rapid validation of kinetic models, but also returns uncertainty-aware predictive models that are valuable tools for model-based decision systems. The platform is tested on a case study of kinetic model identification of complete oxidation of methane on Pd/Al2O3 catalyst, employing a micro packed bed reactor. A suitable kinetic model with precise estimation of its parameters was determined by performing a total of 20 automated experiments, completed in two days

Rapid Screening of Kinetic Models for Methane Total Oxidation using an Automated Gas Phase Catalytic Microreactor Platform

An automated flow micropacked bed catalytic reactor platform was developed to conduct pre-planned experiments for rapid screening of kinetic models. The microreactor was fabricated using photolithography and deep reactive ion etching of a silicon wafer, with a reaction channel width and depth of 2 mm and 420 μm respectively. It was packed with ca. 10 mg of 5 wt. % Pd/Al2O3 catalyst to perform methane combustion, which was the selected reaction to test the developed platform. The experimental system was monitored and controlled by LabVIEW to which Python scripts for online design of experiments and data analysis were integrated. Within each experimental campaign, the platform automatically adjusted the experimental conditions, and the analysis of the product stream was conducted by online gas chromatography. The experimental platform demonstrated the capability of identifying the most probable kinetic models amidst potential models within two days