PDE-Shaped Finite-Relaxation Neural Media for Resolution-Stable Spatial Binding

<title>Abstract</title> <p>We study PDE-shaped finite-relaxation media as neural computationalsubstrates for spatial binding. The model represents trainable spatialcoefficients as continuous conductivity, absorption, and coupling fields, andevaluates them through a fixed number of differentiable relaxation steps. Totest whether such media can support resolution-stable spatial computation, weintroduce a continuous-cross benchmark in which two localized boundary sourcesdefine a hidden crossing coordinate and a teacher field assigns the class atthat coordinate. Across five seeds, the anisotropic medium reaches\((96.13%\pm1.58%)\) accuracy at \((32\times32)\) and maintains\((96.00%\pm1.81%)\) at \((128\times128)\) without retraining in a large held-outaudit. In contrast, a learned generic local-kernel relaxation control fits thetraining resolution but drops to \((37.16%\pm9.71%)\) at \((128\times128)\), and aparameter-matched CoordConv CNN drops to \((54.47%\pm13.53%)\). Spatialshuffle, targeted lesion, interaction-localization, solver-step, and residualanalyses indicate that the learned coefficient fields support an inspectablefinite-relaxation mechanism rather than a fully converged elliptic solver.Strong explicit source-localization baselines solve the task more accurately,so the contribution is not classifier superiority. Instead, the paper providesa controlled computational study of PDE-shaped neural media, their learnedspatial organization, and their resolution-transfer bias.</p>