Interpretable β-Variational Autoencoders for Discovering Higher-Order Sensitivities in Nonlinear Hysteretic Structures

Abstract Hysteretic structural systems exhibit highly nonlinear behaviors governed by parameter interactions, creating challenges for traditional sensitivity analysis methods, especially when higher-order effects dominate. This work introduces a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> -variational autoencoder ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> -VAE) framework that learns low-dimensional, interpretable latent representations of nonlinear hysteretic responses and reveals the dominant parameter interactions that drive the dynamic response behavior. Using simulated response data from a multistory base-isolated building subjected to nonstationary ground motions, we demonstrate that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> -VAE reliably compresses base displacement trajectories into a small number of (linearly) disentangled latent variables that align with physically meaningful generative factors. A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> -annealing strategy is proposed to control latent-space sparsity and sequentially extract salient combinations of generative factors that arise from either structural or excitation parameters. We complement this with multiple linear regression analysis to further identify salient parameter combinations for latent variables and Sobol’ sensitivity analysis to quantitatively rank latent contributions. Through a series of examples that consider uncertainty in the structural and excitation parameters, both separately and jointly, we show that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta" display="inline" overflow="scroll"> <mml:mi>β</mml:mi> </mml:math> -VAE uncovers stiffness–strength axes, duration-frequency controlled excitation modes, and higher-order cross-domain interactions that cannot be easily isolated by conventional Sobol’ indices. These results highlight the potential of VAEs as data-driven yet interpretable tools for nonlinear system analysis, offering a pathway toward model-order reduction, feature discovery, and future physics-integrated learning strategies in structural dynamics.