Statistical Learning Theory for Neural Operators

2026/01/01

paper

Abstract

We present statistical convergence results for the learning of (possibly) non-linear mappings in infinite-dimensional spaces. Specifically, given a map G₀:\mathcal X→\mathcal Y between two separable Hilbert spaces, we analyze the problem of recovering G₀ from n∈ℕ noisy input-output pairs (x_i, y_i)_i=1ⁿ with y_i = G₀ (x_i)+ε_i; here the x_i\inX represent randomly drawn "design" points, and the ε_i are assumed to be either i.i.d. white noise processes or subgaussian random variables in Y. We provide general convergence results for least-squares-type empirical risk minimizers over compact regression classes G⊆ L^∞(X,Y), in terms of their approximation properties and metric entropy bounds, which are derived using empirical process techniques. This generalizes classical results from finite-dimensional nonparametric regression to an infinite-dimensional setting. As a concrete application, we study an encoder-decoder based neural operator architecture termed FrameNet. Assuming G₀ to be holomorphic, we prove algebraic (in the sample size n) convergence rates in this setting, thereby overcoming the curse of dimensionality. To illustrate the wide applicability, as a prototypical example we discuss the learning of the non-linear solution operator to a parametric elliptic partial differential equation.

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