  We introduce a physics-informed Bayesian Neural Network (BNN) with flow
approximated posteriors using multiplicative normalizing flows (MNF) for
detailed uncertainty quantification (UQ) at the physics event-level. Our method
is capable of identifying both heteroskedastic aleatoric and epistemic
uncertainties, providing granular physical insights. Applied to Deep Inelastic
Scattering (DIS) events, our model effectively extracts the kinematic variables
$x$, $Q^2$, and $y$, matching the performance of recent deep learning
regression techniques but with the critical enhancement of event-level UQ. This
detailed description of the underlying uncertainty proves invaluable for
decision-making, especially in tasks like event filtering. It also allows for
the reduction of true inaccuracies without directly accessing the ground truth.
A thorough DIS simulation using the H1 detector at HERA indicates possible
applications for the future EIC. Additionally, this paves the way for related
tasks such as data quality monitoring and anomaly detection. Remarkably, our
approach effectively processes large samples at high rates.
