Quantifying constraint hierarchies in Bayesian PINNs via per-constraint Hessian decomposition

Bayesian physics-informed neural networks (B-PINNs) merge data with governing equations to solve differential equations under uncertainty. However, interpreting uncertainty and overconfidence in B-PINNs requires care due to the poorly understood effects the physical constraints have on the network; overconfidence could reflect warranted precision, enforced by the constraints, rather than miscalibration. Motivated by the need to further clarify how individual physical constraints shape these networks, we introduce a scalable, matrix-free Laplace framework that decomposes the posterior Hessian into contributions from each constraint and provides metrics to quantify their relative influence on the loss landscape. Applied to the Van der Pol equation, our method tracks how constraints sculpt the network's geometry and shows, directly through the Hessian, how changing a single loss weight non-trivially redistributes curvature and effective dominance across the others.