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Bayesian Knowledge Tracing (BKT) is a popular interpretable computational model in the educational mining community that can infer a student's knowledge state and predict future performance based on practice history, enabling tutoring systems to adaptively select exercises to match the student's competency level. Existing BKT implementations do not scale to large datasets and are difficult to extend and improve in terms of prediction accuracy. On the other hand, uninterpretable neural network (NN) student models, such as Deep Knowledge Tracing, enjoy the speed and modeling flexibility of popular computational frameworks (e.g., PyTorch, Tensorflow, etc.), making them easy to develop and extend. To bridge this gap, we develop a collection of BKT recurrent neural network (RNN) cells that are much faster than brute-force implementations and are within an order of magnitude of a fast, fine-tuned but inflexible C++ implementation. We leverage our implementation's modeling flexibility to create two novel extensions of BKT that significantly boost its performance. The first merges item response theory (IRT) and BKT by modeling multidimensional problem difficulties and student abilities without fitting student-specific parameters, allowing the model to easily generalize to new students in a principled way. The second extension discovers the discrete assignment matrix of problems to knowledge components (KCs) via stochastic neural network techniques and supports further guidance via problem input features and an auxiliary loss objective. Both extensions are learned in an end-to-end fashion; that is, problem difficulties, student abilities, and assignments to knowledge components are jointly learned with BKT parameters. In synthetic experiments, the skill discovery model can partially recover the true generating problem-KC assignment matrix while achieving high accuracy, even in some cases where the true KCs are structured unfavorably (interleaving sequences). On a real dataset where problem content is available, the skill discovery model matches BKT with expert-provided skills, despite using fewer KCs. On seven out of eight real-world datasets, our novel extensions achieve prediction performance that is within 0.04 AUC-ROC points of state-of-the-art models. We conclude by showing visualizations of the parameters and inferences to demonstrate the interpretability of our BKT RNN models on a real-life dataset. [ABSTRACT FROM AUTHOR]

A mathematical model for the atmospheric residue desulfurization (ARDS) process was used to estimate the deactivation parameters for the hydrodesulfurization (HDS) and hydrodemetalation (HDM) reactions. These parameters, namely Thiele moduli and contact efficiency, were assessed for three commercial ARDS catalyst systems and three atmospheric residues. The model's deactivation parameters were validated with the literature, even though the model's structure did not explicitly model these free parameters mathematically. It was demonstrated that the Thiele modulus had a positive correlation with the pore volume of the fresh catalyst and a negative correlation with its bulk density for the HDS reaction. The lighter feed, in comparison to the other feeds, contacted the catalysts more efficiently under the current operating conditions. Metal removal depended on the catalyst pore structure, and vanadium removal seemed to be controlled by diffusion. [ABSTRACT FROM AUTHOR]