Expressivity of Programmable-Metasurface-Based Physical Neural Networks: Encoding Non-Linearity, Structural Non-Linearity, and Depth

Abstract

Wave-based signal processing conventionally encodes input data into the input wavefront, making it challenging to implement non-linear operations. Programmable wave systems enable an alternative approach: encoding the input data into the scattering properties of tunable components. With such structural input encoding, two potentially non-linear mappings are involved: first, from the input data to the tunable components' scattering characteristics, and, second, from these scattering characteristics to the output wavefront. In this paper, we systematically examine the expressivity of a wave-based physical neural network (WPNN) with structural input encoding. Our analysis is based on a physics-consistent multiport-network model of a compact D-band rich-scattering cavity parametrized by a 100-element programmable metasurface. We separately control encoding non-linearity, structural non-linearity, and network depth in order to examine their interplay, considering a controlled scalar regression task. With phase encoding and strong inter-element mutual coupling (MC), both aforementioned mappings are strongly non-linear and the WPNN performs very well even with a single layer. We further observe that additional layers can partially compensate for weak inter-element MC. In addition, we demonstrate that WPNN depth can improve expressivity even when it is not associated with an increase in trainable weights. Altogether, our results provide a physics-consistent picture of how encoding choice, MC strength, and depth jointly govern the expressive power of PM-based WPNNs, informing design choices for future experimental implementations of WPNNs.

Figures (7)

• Figure 1: System model for a PM-parametrized cavity. The cavity contains $N_\mathrm{T}$ transmitting antennas, $N_\mathrm{R}$ receiving antennas, and a PM with $N_\mathrm{S}$ tunable PM elements. The $i$th PM element is represented as an antenna element whose port is terminated by a tunable load with reflection coefficient $r_i$. The encoding function maps the $i$th entry of the control vector to $r_i$. The static scattering between the antenna and PM ports is described as a linear network.
• Figure 2: (a) Side view of the D-band PM-parametrized rich-scattering enclosure in the WNoC context of monochristou_toward_2026 (see Table \ref{['tb:stackup']} for further details). (b) Top view of the microstrip patch element employed as antenna element and as PM element (see Table \ref{['tb:patch_config']} for further details). The lumped ports feeding the antenna are depicted in red, and the conductive surfaces in gold. (c) 3D view of the D-band PM-parametrized cavity, with indication of the antenna indices.
• Figure 3: Independent-weights WPNN architecture. Each layer is based on the PM-parametrized cavity modeled in Fig. \ref{['fig:system_model']} whose concrete implementation is shown in Fig. \ref{['fig2']}. Using isolators, the output wavefront from one layer is unidirectionally fed into the next layer as input wavefront. Each layer has its own set of learnable weights, which is emphasized visually by different colors of the loads in each layer. The independent-weights WPNN architecture specializes to the shared-weights WPNN architecture when all layers share the same weights.
• Figure 4: End-to-end channel impulse responses of our PM-parametrized cavity (with $N_\mathrm{T}=N_\mathrm{R}=5$) depicted in Fig. \ref{['fig2']} for three random PM configurations (color-coded). The six vertical lines indicate the six considered finite time-gating truncation times: 0.02 ns, 0.05 ns, 0.1 ns, 0.3 ns, 0.8 ns, and 2.0 ns.
• Figure 5: Non-linear regression performance on 50 target functions obtained with $f_c=0.02$; the curves display the median NMSE across the 50 target functions, and the shades indicate the corresponding standard deviation. (a) Performance dependence on the strength of the structural non-linearity in the single-layer case, for both choices of encoding function (phase encoding and linear encoding); the two WPNN architectures (independent-weights and shared-weights) are identical in the single-layer case. As described in Sec. \ref{['subsec_timegating']}, we control the strength of the structural non-linearity via the truncation time $\tau$ used for time gating that acts as a proxy for the effective MC strength. The horizontal lines correspond to the limit $\tau\rightarrow\infty$ (i.e., no time gating). (b) Performance dependence on the WPNN depth, for both WPNN architectures (independent-weights and shared-weights), both choices of encoding function (phase encoding and linear encoding), and two choices of effective MC strength ($\tau=0.02$ ns and $\tau\rightarrow\infty$).