Quantification of model error for inverse problems in the Weak Neural Variational Inference framework
This work addresses model error quantification for researchers in computational physics and engineering, offering a structured alternative to traditional methods, though it is incremental as it builds on an existing framework.
The paper tackles the problem of biased estimates in PDE-based inverse problems due to unreliable constitutive laws by extending the Weak Neural Variational Inference framework to explicitly quantify model errors, demonstrating improved accuracy and robustness in elastography applications.
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.