This paper presents a deep-learning-based numerical method for pricing barrier options by solving the associated backward stochastic differential equation (BSDE) with boundary conditions. The key innovation is transforming the Cauchy-Dirichlet problem (PDE with boundary conditions) into a standard Cauchy problem by incorporating the barrier boundary condition into the terminal payoff using a Brownian motion bridge. This approach avoids the need for penalty methods or reflected BSDEs used in prior work for American options. The method uses a neural network to approximate the Z_t process (related to the option delta) at each time step, with a merged network structure that takes both the underlying price and time to maturity as inputs. Extensive testing on 72 up-and-out call option cases with varying maturities, volatilities, and barriers shows that the method achieves high accuracy, with average relative errors below 0.6% when batch normalization is applied. The paper also investigates computational efficiency, finding that applying batch normalization only at the input layer yields similar accuracy to full-layer normalization but is approximately three times faster. The results demonstrate that the deep-learning BSDE approach can effectively handle barrier options, providing a flexible alternative to traditional numerical methods.