Multivariate engineering formulas discovery with knowledge-based neural network

Abstract

Multivariate engineering formulas are the foundation of various engineering standards worldwide for constructing complex systems. Traditional formula discovery methods suffer from low efficiency, the curse of dimensionality, and low physical interpretability. To address these limitations, this study proposes a knowledge-based method for efficiently generating multivariate engineering formulas directly from data. The method consists of four components: (1) a deep generative model considering dimensional homogeneity, (2) a physics-adaptive normalization method for multiple engineering variables with different units, (3) a feature merging algorithm grounded in dimensionality theory, and (4) a machine learning-based data segmentation method for piecewise formulas. Experiments on two ground-truth datasets demonstrate that our proposed method improves the accuracy of the generated formulas by 35.6% (measured by mean absolute error), compared to the Eureqa program. Additionally, it enhances the mechanistic interpretability of the results, compared to both Eureqa and the emerging physics-informed neural network-based equation discovery methods. The piecewise formulas successfully capture the implicit mechanisms in the experimental data, consistent with theoretical analysis. Overall, our knowledge-based method holds great promise for improving the efficiency of discovering interpretable and generalizable multivariate engineering formulas, facilitating the transformation of new techniques from testing to applications.