Quantification of model error for inverse problems in the Weak Neural Variational Inference framework

TL;DR

This work extends the Weak Neural Variational Inference framework to quantify model error in PDE-based elastography inverse problems by separating reliable conservation laws from unreliable constitutive laws and treating all state variables as latent. It introduces a virtual likelihood with two residuals, $r^{(e)}$ and $r^{(c)}$, penalized by precisions $(\lambda^{(e)}, \boldsymbol{\lambda}^{(c)})$, enabling identification of regions where the constitutive law breaks down. Variational inference with a structured mean-field posterior and Stochastic Variational Inference is used to learn the posterior over fields $(m, \boldsymbol{u}, \boldsymbol{\sigma})$ and the constitutive-operator discrepancy, providing posterior means and credibility intervals for material properties and stresses. In a synthetic 2D elastography example, the method correctly flags the transverse-isotropic inclusion as invalid and recovers parameters in the valid background, while achieving computational efficiency through weighted residuals and Monte Carlo approximations. The approach offers a principled, interpretable, and scalable route to incorporating uncertainty in constitutive modeling for improved reliability in elastography and related PDE-inverse problems.

Abstract

We present a novel extension of the Weak Neural Variational Inference (WNVI) framework for probabilistic material property estimation that explicitly quantifies model errors in PDE-based inverse problems. Traditional approaches assume the correctness of all governing equations, including potentially unreliable constitutive laws, which can lead to biased estimates and misinterpretations. Our proposed framework addresses this limitation by distinguishing between reliable governing equations, such as conservation laws, and uncertain constitutive relationships. By treating all state variables as latent random variables, we enforce these equations through separate sets of residuals, leveraging a virtual likelihood approach with weighted residuals. This formulation not only identifies regions where constitutive laws break down but also improves robustness against model uncertainties without relying on a fully trustworthy forward model. We demonstrate the effectiveness of our approach in the context of elastography, showing that it provides a structured, interpretable, and computationally efficient alternative to traditional model error correction techniques. Our findings suggest that the proposed framework enhances the accuracy and reliability of material property estimation by offering a principled way to incorporate uncertainty in constitutive modeling.

Figures (6)

• Figure 1: Connection between latents (white circles) and observables (grey boxes).
• Figure 2: Problem setup.
• Figure 3: Expected weighted squared residual $r^{(c)}$ over the iterations ($1$ unit = $10,000$ iterations)
• Figure 4: Displacement fields $u_1$ (left) and $u_2$ (right). On both subfigures: The observations $\hat{\boldsymbol{u}}$ are in the top left, the inferred posterior mean of the displacement fields $\mu_{u_i}$ are in the top right, the absolute error of these two are in the bottom left, and the absolute errors normed by the observation noise $\tau^{-1}$ are the bottom right subplot.
• Figure 5: First line shows the stress means $\boldsymbol{\mu}_{\boldsymbol{\sigma}}$ and second line shows the $95\%$ credibility intervals. In the columns (f.l.t.r.) are shown the components $\sigma_{11}$, $\sigma_{12}$ and $\sigma_{22}$.