Cauchy Activations for Direct Solution of Linear PDEs

Cauchy activations provide an analytic trial space for solving linear partial differential equations by direct least squares. We develop Cauchy Feature Collocation (CFC), which fixes complex poles outside the physical domain and recovers output coefficients from PDE, boundary, and initial constraints. Closed-form mixed derivatives assemble the operator matrices, and a singular-value decomposition determines the coefficients without gradient descent. For targets holomorphic on a neighborhood containing the pole contours, we prove geometric approximation of a fixed finite jet with shared coefficients. Output-specific identities describe recovery from the fitted constraints. Controlled heat and plate experiments show that direct Cauchy fitting achieves lower field and differential errors than coefficient-only Adam under identical matrices and objectives at the tested optimization budget. Comparisons with fixed tanh, sine, Gaussian RBF, and Chebyshev dictionaries using a common direct solver reveal a problem-dependent ordering: Cauchy attains the lowest heat differential error, while sine features lead on plate accuracy. In a separate study of sparse values and slopes across analytic near-singularity families, Cauchy-Complex obtains the lowest aggregate error for held-out higher derivatives.