DeepONet-accelerated Bayesian inversion for moving boundary problems

TL;DR

The paper tackles rapid Bayesian inversion for moving-boundary flows in RTM by replacing costly forward solves with a DeepONet neural-operator surrogate. This surrogate, embedded within an ensemble Kalman inversion framework and augmented with an OfflineUQ discrepancy model, enables real-time, high-resolution inference of spatially varying permeability and porosity, including defect geometry. Synthetic and laboratory tests demonstrate substantial runtime reductions (up to ~200×) with preserved, though slightly inflated, posterior uncertainty and accurate defect localisation. The work also shows generalization across sensor configurations and discusses practical considerations for industrial digital twins, including offline training costs and potential extensions to more complex geometries.

Abstract

This work demonstrates that neural operator learning provides a powerful and flexible framework for building fast, accurate emulators of moving boundary systems, enabling their integration into digital twin platforms. To this end, a Deep Operator Network (DeepONet) architecture is employed to construct an efficient surrogate model for moving boundary problems in single-phase Darcy flow through porous media. The surrogate enables rapid and accurate approximation of complex flow dynamics and is coupled with an Ensemble Kalman Inversion (EKI) algorithm to solve Bayesian inverse problems. The proposed inversion framework is demonstrated by estimating the permeability and porosity of fibre reinforcements for composite materials manufactured via the Resin Transfer Moulding (RTM) process. Using both synthetic and experimental in-process data, the DeepONet surrogate accelerates inversion by several orders of magnitude compared with full-model EKI. This computational efficiency enables real-time, accurate, high-resolution estimation of local variations in permeability, porosity, and other parameters, thereby supporting effective monitoring and control of RTM processes, as well as other applications involving moving boundary flows. Unlike prior approaches for RTM inversion that learn mesh-dependent mappings, the proposed neural operator generalises across spatial and temporal domains, enabling evaluation at arbitrary sensor configurations without retraining, and represents a significant step toward practical industrial deployment of digital twins.

Figures (22)

• Figure 1: Left: realisation of a Gaussian random field used as the level-set function. Right: corresponding thresholded geometry with $L_{*}=1$.
• Figure 2: Configuration of the geometry: central defect region obtained from the level-set function together with the RT regions $\mathcal{RT}_{B}$ and $\mathcal{RT}_{T}$ defined by the random fields $\xi_{B}$ and $\xi_{T}$.
• Figure 3: Five prior samples of log-permeability $\log K^{(j)}$ (top) and porosity $\phi^{(j)}$ (bottom) obtained via Eq. \ref{['eq:param12']}, given $\mathbf{u}_{K,\phi}^{(j)} \sim \mathbb{P}(\mathbf{u}_{K,\phi})$.
• Figure 4: Left: locations of pressure sensors for a dense configuration with $M=100$. Middle: locations of pressure sensors for a sparse configuration with $M=23$, corresponding to the experimental setup. Right: FEM mesh used for the CVFEM simulations.
• Figure 5: Results for the $M=100$ sensor configuration. First column: ground–truth log-permeability, $\log K^{\dagger}$, and porosity, $\phi^{\dagger}$. Second column: prior statistics. Third to fifth columns: posterior statistics obtained with ensemble sizes $J=500$, $J=1000$, and $J=5000$, respectively. From top to bottom: mean of log-permeability, $\overline{\log K}$, mean of porosity, $\overline{\phi}$, standard deviation of log-permeability, $\mathrm{STD}(\log K)$, standard deviation of porosity, $\mathrm{STD}(\phi)$, probability of central defects, $\mathbb{P}_{\mathrm{def}}$, and probability of RT, $\mathbb{P}_{\mathrm{RT}}$. The geometry of the true central defects is shown in white in the fifth row, while the true RT geometry is shown in the bottom row.