Parametric Electromagnetic Information: Field-Manifold Geometry and the Stability of Learned Field Representations
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.