Gradient regularity and potential estimates for fractional drift--diffusion equations in the critical and subcritical ranges
arXiv:2608.19571
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper isolates a scale-critical interaction between fractional diffusion of order 2s and first-order drift, with the rescaled drift coefficient changing as R^{2s-1}. This gives a principled way to make multiscale neural operators resolution-aware: drift-like transport should be attenuated at fine scales when s>1/2, but not when s=1/2. The most transferable construction is a fractional-diffusion residual block whose transport branch is explicitly normalized by this scaling law, with stability tested across image resolutions and model depths.
Ideas from this paper
Unverified
2026
Replace an unconstrained multiscale residual block by the sum of a fractional diffusion branch and a drift or transport branch whose strength follows the PDE scaling law. At finer spatial scales, the drift coefficient is multiplied by R^{2s-1}; this suppresses unstable transport when s>1/2 while preserving equal-strength diffusion and drift at the critical value s=1/2.
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