Regularity for Elliptic Equations with Coefficients of Small Mean Oscillation

What the math gives to ML

The paper provides a constructive stability principle for variable-coefficient second-order operators: uniform ellipticity controls the operator globally, while sufficiently small local mean oscillation allows variable coefficients to be treated as perturbations of constant coefficients. This suggests a neural PDE or implicit feature layer whose learned diffusion tensor is constrained to remain elliptic and locally low-oscillation, giving a substantially more stable spatial operator than unconstrained dynamic filters. The most direct experiment is to replace a standard learned diffusion or convolution block with an implicit elliptic layer and train with a differentiable BMO-style coefficient penalty.