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.

We resort to game theory in order to formulate Data-Driven methods for solid mechanics in which stress and strain players pursue different objectives. The objective of the stress player is to minimize the discrepancy to a material data set, whereas the objective of the strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new non-cooperative Data-Driven games identify an effective material law from the data and reduce to conventional displacement boundary-value problems, which facilitates their practical implementation. However, unlike supervised machine learning methods, the proposed non-cooperative Data-Driven games are unsupervised, ansatz-free and parameter-free. In particular, the effective material law is learned from the data directly, without recourse to regression to a parameterized class of functions such as neural networks. We present analysis that elucidates sufficient conditions for convergence of the Data-Driven solutions with respect to the data. We also present selected examples of implementation and application that demonstrate the range and versatility of the approach.

There are more than a trillion sensors in the world today and according to some estimates there will be about 50 trillion cameras worldwide within the next 5 years, all collecting data either sporadically or around the clock. With such explosive growth of available data and computing resources, recent advances in machine learning and data analytics have yielded transformative results across diverse scientific disciplines, including image recognition, natural language processing, cognitive science, and genomics. However, in many engineering applications, quality and error-free data is not easy to obtain, e.g., for system dynamics characterized by bifurcations and instabilities, hysteresis, delayed responses, and often irreversible responses. Admittedly, as in all everyday applications, in engineering problems, the volume of data has increased substantially compared to even a decade ago but analyzing big data is expensive and time-consuming. Data-driven methods, which have been enabled in the past decade by the availability of sensors, data storage, and computational resources, are taking center stage across many disciplines (physical and information) of science.