The classical dimensional analysis method has limitations in determining the uniqueness and relative importance of the dimensionless quantities. A machine-learning based dimensional analysis method is proposed to address the limitations. The proposed method identifies unique and relevant dimensionless quantities by combining an artificial neural network with the data-driven dimensional analysis. We employ a fully connected neural network to construct the ridge function for the response surface in a physical system. The gradient of the response surface for active subspace analysis is computed based on a finite difference approximation. An effective approach is proposed to determine the independent variables of experimental measurements or numerical simulations for computing the gradient of the response surface. The proposed method is validated by analyzing benchmark pipe flows and a fluid-structure interaction system. The dominant dimensionless quantities obtained by the proposed method are consistent with those reported in the literature. The proposed method has the advantage of identifying the relatively important dimensionless quantities without referring to the complex theoretical equations. (C)& nbsp;2022 Elsevier Inc. All rights reserved.