A Copula-based variational autoencoder for uncertainty quantification in inverse problems: application to damage identification in an offshore wind turbine

TL;DR

A novel Copula-based VAE architecture is proposed that decouples the marginal distribution of the variables from their dependence structure, offering a flexible method for representing complex, correlated posterior distributions in high-dimensional spaces.

Abstract

Structural Health Monitoring of Floating Offshore Wind Turbines (FOWTs) is critical for ensuring operational safety and efficiency. However, identifying damage in components like mooring systems from limited sensor data poses a challenging inverse problem, often characterized by multimodal solutions where various damage states could explain the observed response. To overcome it, we propose a Variational Autoencoder (VAE) architecture, where the encoder approximates the inverse operator, while the decoder approximates the forward. The posterior distribution of the latent space variables is probabilistically modeled, describing the uncertainties in the estimates. This work tackles the limitations of conventional Gaussian Mixtures used within VAEs, which can be either too restrictive or computationally prohibitive for high-dimensional spaces. We propose a novel Copula-based VAE architecture that decouples the marginal distribution of the variables from their dependence structure, offering a flexible method for representing complex, correlated posterior distributions. We provide a comprehensive comparison of three different approaches for approximating the posterior: a Gaussian Mixture with a diagonal covariance matrix, a Gaussian Mixture with a full covariance matrix, and a Gaussian Copula. Our analysis, conducted on a high-fidelity synthetic dataset, demonstrates that the Copula VAE offers a promising and tractable solution in high-dimensional spaces. Although the present work remains in the two-dimensional space, the results suggest efficient scalability to higher dimensions. It achieves superior performance with significantly fewer parameters than the Gaussian Mixture alternatives, whose parametrization grows prohibitively with the dimensionality. The results underscore the potential of Copula-based VAEs as a tool for uncertainty-aware damage identification in FOWT mooring systems.

Figures (9)

• Figure 1: Schematic representation of the system under study. The figure illustrates the two considered damage types: anchoring (displacement of the line anchor) and biofouling (added mass due to mollusks, algae, or other species adhering to the mooring line). Wind and wave incidence follow surge direction, which is perpendicular to the plane of the blades.
• Figure 2: Example of the generated response data of the surge DOF for one scenario in the (a) time and (b) frequency domains.
• Figure 3: correct figure notation for the encoder outputs Variational Autoencoder architecture. The encoder estimates the properties that describe the posterior PDF of the latent space (damaged condition features), $\mathbf{z} \sim q_{\bm{\theta}}(\mathbf{z}|\mathbf{m}, \mathbf{w})$. A sampling layer draws $H$ random samples from the distribution, which are then fed to the optimal forward operator $\mathcal{F}_{\bm{\varphi}^{*}}$. The output layer yields the reconstruction of the input measurements, $\hat{\mathbf{m}}_{i}^{h} = \mathcal{F}_{\bm{\varphi}^{*}} \circ \mathcal{E}_{\bm{\theta}}(\mathbf{m}_{i}^{h}, \mathbf{w}_{i})$, for each sample $h = 1,..., H$.
• Figure 4: Example of ground truth color map in the damaged condition space for an observed measurement $\mathbf{m}_{i}$. According to Algorithm \ref{['alg:algorithm_ground_truth']}, higher density values correspond to lower measurement misfit between $\mathbf{m}_{i}$ and the reconstruction $\hat{\mathbf{m}}_{i} = \mathcal{F}_{\bm{\varphi}^{*}}([\mathbf{z}, \mathbf{w}])$.
• Figure 5: