Identifying parametric expression of probability density function of slowly-varying processes: a data-driven method based on the Π theorem

The response analysis of random vibration systems holds significant importance in comprehending intricate natural phenomena, bolstering the reliability of engineering design, and optimizing structural performance. Traditional analytical approaches frequently encounter hurdles, particularly in deriving parametric expressions for the probability density of system response. This paper aims to propose a novel data-driven methodology based on the Buckingham Π theorem to identify the parametric expression of probability density of slowly-varying processes from state data. The proposed data-driven method is carried out by two successive steps. The first step involves identifying the parametric expressions of invariants of the corresponding conservative system within the framework of the Koopman operator theory in conjunction with the Π theorem. This process yields the expressions for slowly-varying processes. In the second step, the parametric expression of probability density of the slowly-varying processes is identified from the data of slowly-varying processes. This identification is based on the principle of maximum entropy and the Π theorem. The data of slowly-varying processes can be calculated from the random state data with the expression identified in the first step. Sparse optimizations are carried out in both steps. The applicability and effectiveness of the proposed method are illustrated through three typical systems, including the Duffing-van der Pol system, Coulomb friction, and a two-degree-of-freedom system. The extensionality of the parametric expression for the probability density of slowly-varying processes is demonstrated. The proposed data-driven method is capable of reducing system dimensions due to the identified slowly-varying processes with lower dimensions compared to the system state. Additionally, leveraging the Π theorem results in a smaller number of involved dimensionless groups in the identification compared to the original parameters.