Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobolev norms remain poorly quantified, even though these norms control both function values and derivatives and are the natural metrics for PDE well-posedness, stability, and generalization. We develop a functional-analytic framework for operator learning in Sobolev spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical PDE. First, for a continuous nonlinear operator $\mathcal{G}: H^{s}(D)\to H^{t}(D')$ with $s > d/2$ and inputs restricted to a compact subset of $H^{s}(D)$, we prove th...
| Authors | Nicole Hao |
|---|---|
| Primary category | cs.LG |
| Published | 4 May 2026 |
| Last updated at source | 2 August 2026 |
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