AI 中文总结
该研究建立了连接经典逼近理论与衍射光学处理器的统一理论框架，推导了逼近误差界、物理极限与可学习性，为大规模模拟光学计算系统提供了设计原则。
AI 中文摘要
我们提出了一个统一的理论框架，将经典通用逼近理论、傅里叶特征逼近与衍射光学处理器相连接。研究表明，相位编码的衍射处理器可实现有限傅里叶特征展开，其数学完备性源于傅里叶/斯通-魏尔斯特拉斯（Stone-Weierstrass）论证，而其物理可实现性则由优化的空间变化相干点扩散函数（PSF）实现有限系数合成所决定。我们的分析推导了逼近误差界，该误差界区分了傅里叶截断、PSF合成、输入相位误差、光学硬件、读出及噪声的贡献；建立了将逼近复杂度与光学自由度、输入输出空间带宽积关联的缩放关系；推导了由光子统计施加的光子预算和吞吐量极限；为相位量化的衍射函数逼近器构建了有限类别的统计可学习性界；并分析了空间非相干照明的影响。此外，我们还分析了相干光学的级联性，表明通过相干混合和光学读出实现的二次特征扩展提供了一种增强表示的机制，同时与数字神经网络的深度分离结果存在本质区别。我们的分析为衍射非线性函数逼近提供了严谨的理论基础，并确立了数学表达能力、光学硬件资源、统计学习与物理性能极限之间的定量关系，从而为大规模模拟光学计算系统提供了通用设计原则。
英文摘要
We present a unified theoretical framework connecting classical universal approximation theory, Fourier-feature approximation, and diffractive optical processors. We show that phase-encoded diffractive processors implement finite Fourier-feature expansions whose mathematical completeness follows from Fourier/Stone-Weierstrass arguments, while their physical realizability is governed by finite coefficient synthesis through optimized spatially varying coherent point-spread functions (PSFs). Our analyses derive approximation-error bounds that separate Fourier truncation, PSF-synthesis, input phase error, optical hardware, readout, and noise contributions; establish scaling relationships linking approximation complexity to optical degrees of freedom and input/output space-bandwidth products; derive photon-budget and throughput limits imposed by photon statistics; formulate finite-class statistical learnability bounds for phase-quantized diffractive function approximators; and analyze the impact of spatially incoherent illumination. We further analyze coherent optical cascadability and show that quadratic feature expansion through coherent mixing and optical readout provides a mechanism for enhanced representation while remaining fundamentally distinct from the depth-separation results established for digital neural networks. Our analyses provide a rigorous theoretical foundation for diffractive nonlinear function approximation and establish quantitative relationships among mathematical expressivity, optical hardware resources, statistical learning, and physical performance limits, thereby offering general design principles for large-scale analog optical computing systems.
Comments46 Pages, 10 Figures
