Accurate and robust mesh deformation is a key ingredient in unsteady computational fluid dynamics (CFD) simulations involving moving boundaries or deforming domains, where preserving mesh quality is essential to ensure numerical stability and accuracy without frequent remeshing. This work presents a preliminary, physics-informed neural network framework for mesh deformation. The proposed approach combines data-driven learning with physical constraints to predict smooth and consistent deformation fields through a dual neural network architecture with exact boundary conditions enforcement. A dedicated strategy is included to preserve near-wall mesh resolution, which is critical for resolving boundary-layer phenomena in Reynolds-Averaged Navier-Stokes (RANS) simulations. The method is demonstrated on a range of two- and three-dimensional test cases, including configurations with large deformations and multiple moving objects, showing robust behavior and competitive mesh-quality metrics when compared with classical deformation techniques. In addition to accuracy and robustness, particular attention is devoted to computational performance and scalability. A systematic exploration of neural-network hyperparameters is conducted, and mesh-quality indicators such as orthogonality and volume-based metrics are used for quantitative assessment. The impact of batch size on memory consumption and runtime is analyzed on both workstation-level hardware and a multi-GPU cluster environment. Results indicate that increasing the number of batches significantly reduces memory requirements, enabling the treatment of larger meshes, at the cost of a moderate and nearly linear increase in computational time. While classical methods remain more efficient for small-scale problems, the proposed neural approach shows promising scalability for large meshes and complex unsteady simulations. Although still at a preliminary stage, the results suggest that physics-informed learning-based mesh deformation is a viable and flexible alternative, with clear potential for extension to other numerical applications involving PDEs on evolving domains.