Implicit Neural Representation for Multiuser Continuous Aperture Array Beamforming

Implicit neural representations (INRs) can parameterize continuous beamforming functions in continuous aperture arrays (CAPAs) and thus enable efficient online inference. Existing INR-based beamforming methods for CAPAs, however, typically suffer from high training complexity and limited generalizability. To address these issues, we first derive a closed-form expression for the achievable sum rate in multiuser multi-CAPA systems where both the base station and the users are equipped with CAPAs. For sum-rate maximization, we then develop a functional weighted minimum mean-squared error (WMMSE) algorithm by using orthonormal basis expansion to convert the functional optimization into an equivalent parameter optimization problem. Based on this functional WMMSE algorithm, we further propose BeamINR, an INR-based beamforming method implemented with a graph neural network to exploit the permutation-equivariant structure of the optimal beamforming policy; its update equation is designed from the structure of the functional WMMSE iterations. Simulation results show that the functional WMMSE algorithm achieves the highest sum rate at the cost of high online complexity. Compared with baseline INRs, BeamINR substantially reduces inference latency, lowers training complexity, and generalizes better across the number of users and carrier frequency.