Multiobjective evolutionary discovery of equation-based analytical models for dynamical systems

In this article, an approach to modeling dynamical systems in case of unknown governing physical laws has been introduced. The systems of differential equations obtained by means of a data-driven algorithm are taken as the desired models. In this case, the problem of predicting the state of the process is solved by integrating the resulting differential equations. In contrast to classical data-driven approaches to dynamical systems representation, based on the general machine learning methods, the proposed approach is based on the principles, comparable to the analytical equation-based modeling. Models in forms of systems of differential equations, composed as combinations of elementary functions and operation with the structure, were determined by adapted multi-objective evolutionary optimization algorithm. Timeseries describing the state of each element of the dynamic system are used as input data for the algorithm. To ensure the correct operation of the algorithm on data characterizing real-world processes, noise reduction mechanisms are introduced in the algorithm. The use of multicriteria optimization, held in the space of complexity and quality criteria for individual equations of the differential equation system, makes it possible to improve the diversity of proposed candidate solutions and, therefore, to improve the convergence of the algorithm to a model that best represents the dynamics of the process. The output of the algorithm is a set of Pareto-optimal solutions of the optimization problem where each individual of the set corresponds to one system of differential equations. In the course of the work, a library of data-driven modeling of dynamic systems based on differential equation systems was created. The behavior of the algorithm was studied on a synthetic validation dataset describing the state of the hunter-prey dynamic system given by the Lotka-Volterra equations. Finally, a toolset based on the solution of the generated equations was integrated into the algorithm for predicting future system states. The method is applicable to data-driven modeling of arbitrary dynamical systems (e.g. hydrometeorological systems) in cases where the processes can be described using differential equations. Models generated by the algorithm can be used as components of more complex composite models, or in an ensemble of methods as an interpretable component.