LDD-RFM: Learnable Domain Decomposition for Random Feature Models via Variable Projection

Random Feature Models (RFMs) are attractive PDE surrogates because their linear-in-features form reduces training to regularized least squares. Standard RFMs, however, use one stationary spectrum over the whole domain, creating a bandwidth dilemma on heterogeneous PDE solutions: fine scales resolve sharp local gradients but oscillate in smooth regions, while coarse scales miss interface-dominated derivatives. We propose \textbf{LDD-RFM}, a learnable domain-decomposition framework whose local experts remain linear RFMs and whose nonlinear variables specify compact-support partition-of-unity (PoU) gates and local spectral scales. For fixed gates and spectra, expert coefficients are eliminated by ridge regression through variable projection, yielding a differentiable reduced objective with explicit linear-solve and partition-smoothness controls. We prove an $H^1$ error decomposition in which PoU sharpness enters through a product-rule term. Controlled regression and PDE experiments show that LDD-RFM reduces $H^1$ errors by $10\times$ over global and fixed-partition RFM baselines under matched feature budgets; exact-solver, timing, and robustness studies check that these gains are not artifacts of the linear solver or evaluation protocol.