Interior $C^{2,α}$ Regularity for the Quadratic Hessian Equation

What the math gives to ML

The paper identifies the positive branch of the quadratic Hessian operator as a useful nonlinear spectral map: G(A)=sqrt(sigma_2(A)) is elliptic, concave, and positively homogeneous on the cone Gamma_2. These properties can be transferred to coordinate-based neural networks that solve fully nonlinear second-order PDEs, where unconstrained Hessian predictions can leave the elliptic branch and destabilize training. A practical adaptation is to use G(D_x^2 u_theta) as the PDE residual and add a differentiable barrier enforcing the Gamma_2 conditions. The regularity theorem motivates monitoring interior Hessian magnitude and variation, although it does not itself provide a general neural-network training guarantee.