Near-Optimal Learning of Gaussian Sobolev Operators

What the math gives to ML

The paper gives a constructive, data-driven surrogate for operators under Gaussian input measures: PCA first removes low-variance input and output directions, then weighted least squares in a truncated multivariate Hermite basis approximates the latent operator. The transferable asset is the separation of approximation error into PCA truncation, empirical PCA, observation noise, and Sobolev/Hermite truncation terms, with higher Sobolev regularity producing faster spectral convergence. A practical neural-network use is a PCA-compressed Hermite head or residual branch for mappings whose latent coordinates are approximately Gaussian, replacing part of an overparameterized MLP by an explicitly sample-efficient polynomial representation. The resulting model is especially promising for low-dimensional neural operators, surrogate models, and pretrained feature maps where data scarcity makes an unconstrained decoder statistically inefficient.