Solver-Based Fitness Function for the Data-Driven Evolutionary Discovery of Partial Differential Equations

Mikhail Maslyaev, Alexander Hvatov

Abstract: Partial differential equations provide accurate models for many physical processes, although their derivation can be challenging, requiring a fundamental understanding of the modeled system. This challenge can be circumvented with the data-driven algorithms that obtain the governing equation only using observational data. One of the tools commonly used in search of the differential equation is the evolutionary optimization algorithm. In this paper, we seek to improve the existing evolutionary approach to data-driven partial differential equation discovery by introducing a more reliable method of evaluating the quality of proposed structures, based on the inclusion of the automated algorithm of partial differential equations solving. In terms of evolutionary algorithms, we want to check whether the more computationally challenging fitness function represented by the equation solver gives the sufficient resulting solution quality increase with respect to the more simple one. The approach includes a computationally expensive equation solver compared with the baseline method, which utilized equation discrepancy to define the fitness function for a candidate structure in terms of algorithm convergence and required computational resources on the synthetic data obtained from the solution of the Korteweg-de Vries equation.