Nonlocal Tikhonov Regularization: Hilbert Scales, Explicit Rates, and the Classical Limit

What the math gives to ML

The paper provides a transferable Hilbert-scale construction: a fractional Sobolev penalty is represented by the positive self-adjoint operator A_s = I + (-Delta)^s, and the inverse problem becomes ordinary Tikhonov regularization after the isometric variable change v = A_s^(1/2)u. The useful neural-network asset is a principled frequency-selective regularizer and preconditioner whose strength is controlled continuously by the fractional order s. This can be inserted into neural fields, image-reconstruction CNNs, or learned PDE solvers whose outputs live on a spatial grid, with FFT or sine transforms making the operator practical. The backward-heat spectral identity additionally gives a concrete diagnostic for suppressing frequencies that are unrecoverable under a smoothing forward map.