Ljupčo Todorovski (2003) Using domain knowledge for automated modelling of dynamic systems with equation discovery. PhD thesis.
The process of establishing an acceptable model of an observed dynamic system from measured data is a challenging task that occupies a major portion of the work of the mathematical modeler. In this thesis, we propose a knowledge-based approach to automated modeling of dynamic systems based on equation discovery methods. Most work in equation discovery is concerned with assisting the empirical approach to modeling physical systems. Following this approach, the observed system is modeled on a trial-and-error basis to fit observed data. None of the available domain knowledge about the observed system (or a very limited portion thereof) is used in the modeling process. The empirical approach contrasts with the theoretical approach to modeling, in which the basic physical processes involved in the observed system are first identified. A human expert then uses domain knowledge about the identified processes to write down a proper structure of the model equations. The equation discovery methods presented in the thesis deal with the problem of integrating the theoretical and empirical approaches to modeling of dynamic systems by integrating different types of theoretical knowledge in the discovery process. Two different types of domain-specific modeling knowledge are considered herein. The first concerns basic processes that govern the behavior of systems in the observed domain. The second concerns existing models that are already established in the domain. In addition, the scope of the existing equation discovery methods is extended toward the discovery of partial differential equations that are capable of modeling both temporal and spatial changes of the state of the observed system. The newly developed methods are successfully applied to different tasks of modeling real-world systems from artificial and real measurement data in the domains of population dynamics, neurophysiology, classical mechanics, hydrodynamics, and Earth science.
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