The emergence of Physics-Informed Neural Networks has introduced a novel computational approach for data assimilation. Although quantifying epistemic and aleatoric uncertainties efficiently for complex and high dimensional problems remains an open question. In this work, we propose a novel method for computing global uncertainties in PDEs governing conservation laws within a Bayesian framework, by combining local Bayesian Physics-Informed Neural Networks (B-PINN) with domain decomposition. The continuity of solutions is obtained by imposing the flux continuity across the interfaces of subdomain. To demonstrate the effectiveness of the proposed methods, we conducted a series of computational experiments on PDEs in 1D and 2D spatial domains. Although we have adopted conservative PINNs in this work, the method can be seamlessly extended to other domain decomposition techniques, such as XPINN, Finite-basis PINN, hp-VPINN and others. The outcome of computational experiments infers that we proposed method recovers the global uncertainty by computing the local uncertainty exactly but more efficiently as the uncertainty in each subdomain can be computed concurrently. The robustness of method is verified by adding the random noise to initial and boundary conditions up to 15 %. The demonstrated method is tested for several types of PDEs using a set of experiments to illustrate the behavior and convergence for different types of problems. The following test are performed: • evaluation over the different types of conservative PDEs (Allen-Cahn, Fokker-Planck, Burgers’ and Korteweg–De Vries) • performance for non-conservative PDE using Fisher-KPP equation • performance evaluation given different levels of added noise: 0%, 5%, 10% and 15% • test on model response to real data (measurements) vs synthetic data • ability to handle subdomains with different levels of added noise • ability to handle subdomains of different sizes with different neural network architectures • test both forwards and inverse formulation of PINN (equation solution and equation discovery) • model’s performance for 1D and 2D multi-scale problems. The results show sufficiently stable performance of B-PINN with domain decomposition for a variety of PDEs with different levels of added noise. Interestingly, while traditional conservative PINNs or cPINNs are not able to handle non-conservation PDEs, using B-PINN allows modifying cPINN to work for non-conservation laws.