Physics-constrained mixture density networks for the inverse problem of Risley prism beam steering

Risley prism scanners require fast inverse mapping from screen coordinates to prism angles, yet the mapping is non-unique and unstable under non-paraxial, thick-prism conditions. This paper presents a physics-constrained mixture density network for the inverse problem of Risley prism beam steering, enabling direct prediction of both prism rotation angles from screen coordinates without paraxial assumptions or auxiliary exit-ray sensing. The model learns a multi-modal von Mises mixture and embeds a high-precision forward ray model in the loss, followed by one-step GN/LM refinement. Simulations demonstrate that, over the main region of the screen, the method achieves >94% correct-solution identification, micrometer-level reprojection RMSE, and millisecond-level inference with improved boundary and moderate-noise robustness. These results support the feasibility of the proposed method for real-time, high-precision pointing control in rotating double-prism systems.