Mathematical modelling is an indispensable tool to support water resource recovery facility (WRRF) operators and engineers with the ambition of creating a truly circular economy and assuring a sustainable future. Despite the successful application of mechanistic models in the water sector, they show some important limitations and do not fully profit from the increasing digitalisation of systems and processes. Recent advances in data-driven methods have provided options for harnessing the power of Industry 4.0, but they are often limited by the lack of interpretability and extrapolation capabilities. Hybrid modelling (HM) combines these two modelling paradigms and aims to leverage both the rapidly increasing volumes of data collected, as well as the continued pursuit of greater process understanding. Despite the potential of HM in a sector that is undergoing a significant digital and cultural transformation, the application of hybrid models remains vague. This article presents an overview of HM methodologies applied to WRRF and aims to stimulate the wider adoption and development of HM. We also highlight challenges and research needs for HM design and architecture, good modelling practice, data assurance, and software compatibility. HM is a paradigm for WRRF modelling to transition towards a more resource-efficient, resilient, and sustainable future.
First-principles modeling of dynamical systems is a cornerstone of science and engineering and has enabled rapid development and improvement of key technologies such as chemical reactors, electrical circuits, and communication networks. In various disciplines, scientists structure the available domain knowledge into a system of differential equations. When designed, calibrated, and validated appropriately, these equations are used to analyze and predict the dynamics of the system. However, perfect knowledge is usually not accessible in real-world problems. The incorporated knowledge thus is a simplification of the real system and is limited by the underlying assumptions. This limits the extent to which the model reflects reality. The resulting lack of predictive power severely hampers the application potential of such models. Here we introduce a framework that incorporates machine learning into existing firstprinciples modeling. The machine learning model fills in the knowledge gaps of the first-principles model, capturing the unmodeled dynamics and thus improving the representativeness of the model. Moreover, we show that this approach lowers the data requirements, both in quantity and quality, and improves the generalization ability in comparison with a purely data-driven approach. This approach can be applied to any first-principles model with sufficient data available and has tremendous potential in many fields.INDEX TERMS differential equations, dynamical systems, first-principles modeling, hybrid modeling, machine learning
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