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

2025-02-11

AI Summary

In PDE-based inverse problems, unreliable constitutive relations induce systematic bias in material parameter estimation. To address this, we propose Weak Neural Variational Inference (WNVI), the first framework to explicitly decouple reliable conservation laws from uncertain constitutive relations. WNVI introduces a weighted-residual virtual likelihood to interpretably localize model error sources and enable probabilistic quantification and robust correction of systematic modeling errors. Integrating weak-form neural networks, variational inference, and latent-variable probabilistic modeling, the method operates without an exact forward solver, ensuring computational efficiency and structural clarity. Evaluated on elastography, WNVI significantly improves both accuracy in material parameter estimation and reliability in uncertainty calibration. It establishes an interpretable, generalizable uncertainty modeling paradigm for PDE-driven inverse problems.

Technology Category

Reasoning under Uncertainty: Probabilistic InferenceMachine Learning: Calibration & Uncertainty QuantificationCognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructuresGraph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsUser Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Quantify model errors in PDE-based inverse problems.
• Distinguish reliable equations from uncertain constitutive laws.
• Enhance accuracy and reliability in material property estimation.

Innovation

• Weak Neural Variational Inference
• virtual likelihood approach
• uncertain constitutive relationships