Modeling complex multiphysics systems, such as tumor growth in soft biphasic tissues, presents significant challenges due to the nonlinear, high-dimensional nature of coupled processes involved. This work introduces a novel AI-accelerated computational scheme for efficiently solving such multiphysics problems by integrating an AI-based surrogate prediction framework within a partitioned (staggered) solver. The scheme specifically addresses the demands of a porous hyperelastic growth model coupled with multiple advection-diffusion equations, which collectively simulate the biomechanical and biochemical interactions in tumor growth within soft tissue environments. To expedite the computational process while maintaining accuracy, the solution approach employs an AI-driven surrogate model based on a combination of feedforward neural networks (FFNN) and autoencoders. Each physical component of the model—representing distinct yet interacting processes—is supported by a tailored surrogate that predicts approximate solutions, reducing the burden on traditional solvers and enhancing scalability on high-performance computing (HPC) platforms. The surrogate-driven partitioned framework not only improves computational efficiency but also maintains robustness across the nonlinear interactions, characteristic of multiphysics models in biological tissue applications. This study highlights the potential of hybrid AI-HPC schemes in tackling the computational demands of high-fidelity multiphysics simulations, underscoring AI's role in enabling real-time solution updates and adaptive scalability in dynamic simulation environments. These findings contribute to the broader objectives of integrating AI with HPC to push the frontiers of computational mechanics, providing a pathway for future advancements in biomedical simulation and other fields involving complex, large-scale multiphysics problems.