Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

ArXi:2605.08170v1 Announce Type: new Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces. However, their approximation properties in Sobole 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 Sobole spaces and connect it to the numerical behavior of Fourier Neural Operators (FNOs) on a prototypical