Residual-Christoffel Sampling for Random Feature Collocation of Linear PDEs

What the math gives to ML

The paper provides an operator-aware way to stabilize fixed-feature PDE solvers: sample points according to leverage in the space of operator-applied features rather than according to the physical domain measure. The key transferable asset is that residual-space whitening makes the expected collocation Gram approximately isotropic, while inverse-density weights prevent high-leverage regions from dominating the least-squares system. This can be brought directly into random-feature MLPs, physics-informed extreme learning machines, or linearized neural PDE solvers without changing the forward architecture. The most informative first test is whether residual-Christoffel sampling reduces condition numbers and reaches a target PDE error with fewer collocation points than uniform sampling or standard residual-adaptive sampling.