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Physics-Informed Neural Networks (PINNs) are gaining popularity as a methodfor solving differential equations. While being more feasible in some contextsthan the classical numerical techniques, PINNs still lack credibility. A remedyfor that can be found in Uncertainty Quantification (UQ) which is justbeginning to emerge in the context of PINNs. Assessing how well the trainedPINN complies with imposed differential equation is the key to tacklinguncertainty, yet there is lack of comprehensive methodology for this task. Wepropose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates thediscrepancy between the B-PINN solution and the unknown true solution. Weexploit recent results on error bounds for PINNs on linear dynamical systemsand demonstrate the predictive uncertainty on a class of linear ODEs.

Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks