The authors develop approximation-theory bounds for neural operators (networks that learn mappings between function spaces) measured in Sobolev norms—norms that penalize both function values and derivatives. They prove that approximating a continuous nonlinear operator to ε error in Sobolev space requires O(ε^(−d/s)) parameters, then validate this scaling on Fourier Neural Operators trained to solve the 1D Burgers PDE. Empirical log-log plots show FNO test error decreasing as a power law in parameter count, with an exponent close to the theoretical prediction.
Main takeaways:
- Sobolev norms track both function values and derivatives, making them the natural metric for PDE solution operators and generalization.
- The paper proves a complexity–error relation: approximating an operator in Sobolev space to error ε requires roughly ε^(−d/s) parameters, where d is dimension and s is smoothness.
- Training FNOs on the Burgers equation with an H¹ loss achieves test errors down to 10⁻⁷ and relative errors ~10⁻³, with both solutions and derivatives matching held-out data.
- Empirically, Sobolev error vs. parameter count follows a power law with exponent ~1.4, reasonably consistent with the theory.