Empirical-Bayes XTFC for Inverse Parameter Estimation

We introduce Empirical-Bayes XTFC for inverse parameter estimation in boundary-value problems. The method embeds Dirichlet boundary conditions exactly via the Extreme Theory of Functional Connections (XTFC) and combines noisy measurements with physics residuals in a lightweight linear model built from fixed Gaussian RBF basis functions. We target the PDE parameter $\nu$ and select it by empirical Bayes: for each candidate $\nu$ we maximize the marginal likelihood (evidence) with respect to a single prior precision $\eta$ placed on the XTFC weights (the linear coefficients in the constrained RBF basis), then choose the $\hat{\nu}$ achieving the highest evidence. This yields posterior means and uncertainty bands without iterative PDE solves or boundary-penalty tuning. On a 1D convection--diffusion benchmark with sharp boundary layers, the approach accurately recovers the diffusion coefficient $\nu$ from only 50 measurements with modest compute, and remains robust in the small-$\nu$ regime where identifiability is challenging. The formulation is simple, stable, and easy to implement. For reproducibility, our source code is available at \url{https://github.com/vikas-dwivedi-2022/OPT_Bayesian_XTFC}