Parameter estimation of structural dynamics with neural operators enabled surrogate modeling

TL;DR

The paper tackles parameter estimation in structural dynamics by introducing a unified deep learning framework that combines parameterization, a neural-operator forward surrogate, and inverse modeling via gradient-based initialization plus neural refinement. A key component is Parametric DeepONet, which encodes system parameters, time via Fourier features, and a choice of linear or nonlinear decoding to predict responses under varying excitations and parameters. The approach is validated on a numerical Duffing oscillator and an experimental wind turbine blade, demonstrating accurate forward predictions and robust inverse estimates in both interpolation and extrapolation settings, with neural refinement reducing ill-posedness. The framework offers a flexible, differentiable, and scalable pathway for surrogate modeling, structural identification, damage detection, and inverse design in structural dynamics.

Abstract

Parameter estimation in structural dynamics generally involves inferring the values of physical, geometric, or even customized parameters based on first principles or expert knowledge, which is challenging for complex structural systems. In this work, we present a unified deep learning-based framework for parameterization, forward modeling, and inverse modeling of structural dynamics. The parameterization is flexible and can be user-defined, including physical and/or non-physical (customized) parameters. In the forward modeling, we train a neural operator for response prediction -- forming a surrogate model, which leverages the defined system parameters and excitation forces as inputs to the model. The inverse modeling focuses on estimating system parameters. In particular, the learned forward surrogate model (which is differentiable) is utilized for preliminary parameter estimation via gradient-based optimization; to further boost the parameter estimation, we introduce a neural refinement method to mitigate ill-posed problems, which often occur in the former. The framework's effectiveness is verified numerically and experimentally, in both interpolation and extrapolation cases, indicating its capability to capture intrinsic dynamics of structural systems from both forward and inverse perspectives. Moreover, the framework's flexibility is expected to support a wide range of applications, including surrogate modeling, structural identification, damage detection, and inverse design of structural systems.

Figures (21)

• Figure 1: Structural dynamics is defined by the mapping (parameterized by system parameters) from excitation forces to dynamic responses.
• Figure 2: Illustrations of vanilla DeepONet and Parametric DeepONet.
• Figure 3: A unified framework for forward and inverse modeling. In forward modeling, a neural network takes excitation force $f$ and system parameter $\boldsymbol{\mu}$ as input and predicts dynamic response $y$. In inverse modeling, gradient-based initialization initializes system parameters $\boldsymbol{\mu}$ by minimizing the forward prediction loss given the $(f, y)$; subsequent neural refinement employs a neural network to take the initially estimated parameter as input and generates refined parameter estimation results. The standard deviation and mean value are computed based on parameter estimation with different initializations. A detailed workflow is further explained in Algorithm \ref{['algo_forward']} and \ref{['algo2']}.
• Figure 4: Ranges of system parameters of training and test data in Case 1 ($\mu_1$ is the stiffness, $\mu_2$ is the damping).
• Figure 5: A data sample in Case 1. Left: the excitation force; Right: the response acceleration ($f(t), \ddot{x} - m/s^2 ,t - s$ ).