Functional-Hybrid modeling through automated adaptive symbolic regression for interpretable mathematical expressions

Mathematical models used for the representation of (bio)-chemical processes can be grouped into two broad paradigms: white-box or mechanistic models, completely based on knowledgeor black-box data-driven models based on patterns observed in data. However, in the past two-decade, hybrid modeling that explores the synergy between the two paradigms has emerged as a pragmatic compromise. The data-driven part of these has been largely based on conventional machine learning algorithms (e.g., artificial neural network, support vector regression), which prevents interpretability of the finally learnt model by the domain experts. In this work, we present a novel hybrid modeling framework, the Functional-Hybrid model, that uses the ranked domain-specific functional beliefs together with symbolic regression to develop dynamic models. We demonstrate the successful implementation of the Functional-Hybrid model and its interpretability, focusing on applying chemical reaction kinetic principles to classical chemical reactions, biochemistry, ecology, physiology, and a bioreactor. Furthermore, we demonstrate that during interpolation, the Functional-Hybrid model performs similarly to a HybridANN hybrid model implementing a conventional ANN. However, it provides the advantage of being to some extent-interpretable, unlike the conventional Hybrid-ANN model. Additionally, it is shown that the Functional-Hybrid model outperforms the Hybrid-ANN model for a very low number of experiments, making it more suitable when data is scarce. Finally, the Functional-Hybrid models show superior extrapolation capabilities compared to the Hybrid-ANN model. This improved performance can be attributed to the structure imposed by the functional transformations introduced in the Functional-Hybrid model.