Maximum a Posteriori Estimation for Linear Structural Dynamics Models Using Bayesian Optimization with Rational Polynomial Chaos Expansions

TL;DR

This work targets efficient MAP estimation for Bayesian updating of linear structural dynamics using frequency-domain data. It introduces Rational Polynomial Chaos Expansions (RPCE) as fast surrogates for FRFs, and develops a sparse Bayesian learning framework with a Laplace approximation to learn RPCE coefficients and quantify surrogate uncertainty. A Bayesian-optimization–driven active design then sequentially enriches the experimental design by maximizing expected improvement, reducing the number of expensive model evaluations. The methodology is demonstrated on a two-degree-of-freedom system and a finite-element model of a cross-laminated timber plate, showing superior MAP accuracy and reduced computational burden compared with fixed-design RPCE, and highlighting practical implications for reliable structural model updating under parameter uncertainty.

Abstract

Bayesian analysis enables combining prior knowledge with measurement data to learn model parameters. Commonly, one resorts to computing the maximum a posteriori (MAP) estimate, when only a point estimate of the parameters is of interest. We apply MAP estimation in the context of structural dynamic models, where the system response can be described by the frequency response function. To alleviate high computational demands from repeated expensive model calls, we utilize a rational polynomial chaos expansion (RPCE) surrogate model that expresses the system frequency response as a rational of two polynomials with complex coefficients. We propose an extension to an existing sparse Bayesian learning approach for RPCE based on Laplace's approximation for the posterior distribution of the denominator coefficients. Furthermore, we introduce a Bayesian optimization approach, which allows to adaptively enrich the experimental design throughout the optimization process of MAP estimation. Thereby, we utilize the expected improvement acquisition function as a means to identify sample points in the input space that are possibly associated with large objective function values. The acquisition function is estimated through Monte Carlo sampling based on the posterior distribution of the expansion coefficients identified in the sparse Bayesian learning process. By combining the sparsity-inducing learning procedure with the sequential experimental design, we effectively reduce the number of model evaluations in the MAP estimation problem. We demonstrate the applicability of the presented methods on the parameter updating problem of an algebraic two-degree-of-freedom system and the finite element model of a cross-laminated timber plate.

Figures (9)

• Figure 1: Illustration of data and model fusion through Bayes' rule. Data is gathered on a structure at specific locations. We consider this data in the frequency domain through application of the Fourier transform (denoted by $\mathcal{F}$) to the time domain data. The system response is also predicted through a simulation model. The information from observations as well as the models can then be combined through Bayes' rule to update the model parameters in $\mathbf{X}$.
• Figure 2: Illustration of the model error in the complex plane that is defined to derive the likelihood function. We assume an additive error $\Tilde{\varepsilon}_k$ between the product of original model response and denominator polynomial $Q(\mathbf{x}^{(k)}) \mathcal{M} (\mathbf{x}^{(k)})$ and the numerator polynomial $P(\mathbf{x}^{(k)})$.
• Figure 3: Illustration of the inference procedure using a surrogate model. The solid line represents the original model $\mathcal{M}(\mathbf{x})$. Based on the prior distribution $f(x)$, a set of input samples $\{x^{(k)}\}$, the experimental design, is generated. The corresponding model output samples, the training set, is computed as $\mathcal{M}(x^{(k)})$. The trained surrogate model $\hat{\mathcal{S}}(x^{(k)})$ can usually be expected to show good accuracy in the bulk of the prior distribution. For the inverse relationship, the model output and the observed data are related via the model error $\varepsilon_{\mathcal{M}}$ and the additive observation noise $\varepsilon_{\mathcal{O}}$. If the surrogate model is used in the inverse problem, an additional random error $\varepsilon_{\mathcal{S}}$ can be considered. If now data $\mathcal{D}_{\mathcal{O}}$ is observed significantly distant from the training samples, the surrogate model accuracy in the corresponding region in the input space might be poor. Thus, the inferred samples, or the MAP estimate, might be prone to significant error. This motivates the development of the active learning procedure to place experimental design samples towards the bulk of the posterior distribution. Remark: We disregard the probabilistic nature of the Bayesian regression surrogate model in this illustration for reasons of simplicity. The surrogate model response is really a random process over $x$.
• Figure 4: Illustration of the posterior sampling and the hierarchical dependency.
• Figure 5: Illustration of two-degree-of-freedom system with fully dependent spring, mass and damping coefficient. The acceleration $\ddot u$ is measured at the second mass, while the frequency response is considered with respect to the first mass with forcing $f$. The frequency response is then $h_{21} (\omega) = \frac{\Tilde{\ddot u} (\omega)}{\Tilde{f} (\omega)} = - \omega^2\frac{\Tilde{u} (\omega)}{\Tilde{f} (\omega)}$ (compare with Eq. \ref{['eq:2dof_appl_frf']}).