A muzzle brake is a mechanical component mounted at the end of a barrel that redirects the flow of propellant gases during the after-effects period of firing in order to reduce recoil forces transmitted to the cradle. This function is especially crucial for large-caliber howitzers, as it mitigates the impact of firing on other mechanical components, thereby reducing weight and increasing battlefield mobility. However, improvements in recoil reduction involve trade-offs, resulting in increased impulse noise and elevated shock-wave intensity. In this study, a physics-informed neural network (PINN)-based surrogate model is proposed to address the multi-objective optimization of large-caliber howitzer muzzle brakes. By balancing recoil mitigation against impulse noise and shock-wave effects, the framework enables systematic exploration of performance trade-offs. The training data for the deep-learning model are generated using computational fluid dynamics (CFD) and computational aeroacoustics (CAA) simulations. By embedding physical constraints into the learning process, the proposed PINN surrogate promotes physical consistency and improved generalization compared with purely data-driven models. The proposed methodology is designed to reduce dependence on extensive high-fidelity simulations while maintaining physically admissible predictions across multiple objectives. The optimized muzzle brake designs obtained from the framework are qualitatively consistent with expected physical trends when compared to the baseline configuration. This research highlights the potential of physics-informed neural surrogate modeling as a methodologically robust alternative to traditional surrogate methods for complex multi-objective optimization problems.