Fourier ptychographic enhancement of iterative pathways: autonomous 3D momentum coordination in hybrid ML-PIE architectures

While data-driven deep learning has empirically advanced ptychographic reconstruction, its inherent limitations-a lack of theoretical interpretability and limited adaptability-remain unresolved. Emerging hybrid architectures integrate physics-based ptychographic iterative engines (PIE) with machine-learning (ML) optimization to preserve interpretability while achieving superior gradient-search performance. Our previous work introduced momentum-accelerated co-optimization (using first- and second-order methods) for single-iteration PIE updates, which simplified hyperparameter configuration in ML-enhanced modules. However, PIE's inherent process of two-dimensional fixed-point adjustment creates a paradox between optimization and stability: achieving high performance requires compensatory hyperparameters to balance transient performance and long-term convergence. This dilemma leads to a fundamental conflict between momentum-driven adaptability and iterative equilibrium, posing a challenge for developing universally stable hybrid architectures. To address these limitations, we have revisited the optimization direction selection in conventional PIE workflows by analyzing Fourier ptychographic microscopy (FPM). We introduce a three-dimensional (3D) autonomous iterative path design framework in which the reconstruction stage is treated as a third spatial dimension. This transforms the conventional challenge of 2D fixed-point tuning into a systematic parameter space planning problem. Extensive tests demonstrate that our proposed method, Adam-DPIE (Dynamic PIE with Adaptive Moment Estimation integration), overcomes three key constraints in current designs: the large number of hyperparameters, hyperparameter sensitivity, and the trade-off between optimization and stability. Remarkably, Adam-DPIE achieves this with only a single hyperparameter while maintaining backward compatibility. This approach provides both methodological insights into PIE research and practical solutions enabling high-performance biomedical imaging systems.