参数化电磁信息：场流形几何与学习到场表示的稳定性
Parametric Electromagnetic Information: Field-Manifold Geometry and the Stability of Learned Field Representations
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中文总结 AI 辅助
该研究拓展电磁信息论至参数化场族，区分电磁本征维度与度量拉伸指数，推导稳定场表示的下界，通过物理约束自编码器验证，揭示同本征维度系统的度量增长差异及解码器灵敏度的线性缩放规律。
中文摘要 AI 辅助
在许多电磁系统中，辐射场或散射场的集合仅由少数物理参数控制，因此形成了嵌入高维观测空间的低维流形。本文将电磁信息论（Electromagnetic Information Theory, EIT）拓展至这类参数化场族，区分了经典线性自由度（NDF）分析中合并的两个描述量：电磁本征维度（Electromagnetic Intrinsic Dimension, EID）$d$，用于计数场变化的局部独立方向；以及度量拉伸指数$ν$，用于支配本征度量体积的电尺寸缩放。利用柯尔莫哥洛夫$\boldsymbol{\u03b5}$熵和$\boldsymbol{\u03b5}$容量，我们推导出下界，表明稳定的非线性表示不仅依赖于维度，还依赖于度量体积的增长，后者会以解码器灵敏度负担的形式再次出现。在加性高斯噪声下，拉回度量与费希尔信息矩阵（Fisher Information Matrix）成正比，将同一几何结构与克拉美罗（Cramér–Rao）估计界关联起来。物理约束自编码器可对达到指定重建精度所需的潜维度给出可操作估计，并为相关的归一化解码器灵敏度提供经验代理。阵列和散射基准测试表明，具有相同本征维度的系统会因物理调制或散射机制的不同而呈现出不同的度量增长规律；而专门的转向弧实验提供了有限样本证据，即观测到的最佳解码器灵敏度代理在电孔径上几乎随度量长度线性缩放，与预测的下界趋势一致。
英文摘要
In many electromagnetic systems, the set of radiated or scattered fields is controlled by only a few physical parameters and therefore forms a low-dimensional manifold embedded in a high-dimensional observation space. This paper extends Electromagnetic Information Theory (EIT) to such parametric field families by separating two descriptors that classical linear NDF analysis merges: the Electromagnetic Intrinsic Dimension (EID), $d$, which counts the locally independent directions of field variation, and the Metric Stretching Exponent, $ν$, which governs the electrical-size scaling of the intrinsic metric volume. Using Kolmogorov $\varepsilon$-entropy and $\varepsilon$-capacity, we derive lower bounds showing that stable non-linear representations depend not only on dimension but also on metric-volume growth, which reappears as a decoder-sensitivity burden. Under additive Gaussian noise, the pullback metric is proportional to the Fisher Information Matrix, linking the same geometry to Cramér--Rao estimation bounds. Physics-constrained autoencoders provide an operational estimate of the latent dimension required to achieve a prescribed reconstruction accuracy and an empirical proxy for the associated normalized decoder sensitivity. Array and scattering benchmarks show that systems with the same intrinsic dimension can exhibit different metric-growth laws depending on the physical modulation or scattering regime, while a dedicated steering-arc experiment provides finite-sample evidence that the best observed decoder-sensitivity proxy scales nearly linearly with metric length across electrical apertures, consistently with the predicted lower-bound trend.
