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 knowledge or 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 have been largely based on conventional machine learning algorithm (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 these hybrid models for four benchmark systems and a microbial fermentation reactor, all of which are systems of (bio)chemical relevance. We also demonstrate that compared to a similar implementation with the conventional ANN, the performance of Functional-Hybrid model is at least two times better in interpolation and extrapolation. Additionally, the proposed framework can learn the dynamics in 50% lower number of experiments. This improved performance can be attributed to the structure imposed by the functional transformations introduced in the Functional-Hybrid model.