<p>The acoustic holography inverse problem aims to reconstruct the wavefront of a sound field by recovering the corresponding phase distribution from amplitude measurements in the target plane. To address slow convergence and limited stability in traditional iterative algorithms (e.g., Gerchberg-Saxton, IASA), this paper derives the forward and inverse projection operators of the sound field from a signal-processing and mathematical-modeling perspective based on the angular spectrum method (ASM). The holographic inversion is formulated as a signal recovery problem involving these projection operators. In the supervised stage, we construct an initial amplitude-phase mapping using contrastive loss and topological smoothness regularization. A physical consistency loss is then introduced in the self-supervised stage to refine the prediction results. Simulation experiments on a controlled MNIST-derived ASM benchmark show that the proposed method achieves faster inference than the implemented iterative baselines while maintaining competitive reconstruction accuracy in terms of MSE, PSNR, and SSIM. It also shows stable performance on the held-out simulated benchmark and under the tested additive noise perturbations. The proposed framework integrates mathematical modeling with data-driven inference, offering an interpretable approach for future real-time sound-field reconstruction studies.</p>