Cubature from rational approximation

What the math gives to ML

The paper gives a constructive way to turn boundary samples into a small set of interior integration nodes and weights: apply the Cauchy–Green identity, rationally approximate the boundary antiderivative, and use the rational poles and residues as cubature data. The transferable asset is not merely rational interpolation, but the ability to replace a dense two-dimensional integral by a geometry-dependent sparse evaluation rule with an observable boundary residual. This is promising for neural fields and neural operators whose spatial features are analytic or approximately analytic, where global pooling, loss integration, or latent aggregation can be reduced to a few interior evaluations. The first implementation should target a fixed domain geometry and compare rational-pole cubature against tensor-product and Monte Carlo integration at equal function-evaluation cost.