Conservative Density-Functional Network

Use a fully convolutional scalar-output network Phi_theta that maps a complete periodic or padded density profile rho to one scalar. To preserve extensivity, sum local contributions: Phi_theta[rho]=sum_r q_theta(rho)_r, with q produced by a cubic-equivariant CNN; then call reverse-mode automatic differentiation to obtain c_hat=-grad(Phi,rho). Train with Monte Carlo targets c_target using an L2 loss, optionally add a free-energy anchor loss on known homogeneous states and a Hessian reciprocity penalty only if numerical differentiation reveals implementation errors. Pseudocode is: rho.requires_grad_(True); Phi=network(rho).sum(); c_hat=-autograd.grad(Phi,rho,create_graph=True); loss=mean((c_hat-c_target)^2); update theta. At inference, solve the Euler-Lagrange fixed point with damped updates above and estimate the largest Jacobian eigenvalue by 10 to 20 power iterations using Jacobian-vector products; set alpha=min(alpha_max,0.8/lambda_max). The paper supplies the thermodynamic relation between c^(1) and the free-energy functional; network values, target noise, and lambda_max are estimated. First experiment: train on 3D lattice hard-cube Monte Carlo profiles in randomized external potentials, comparing direct c prediction with the conservative model at equal parameter count. Test homogeneous equations of state, planar-wall profiles, and random initial densities. The prediction is that the reciprocity residual ||J-J^T||_F/||J||_F is below 1e-4 for the conservative model but remains measurable for direct prediction, and fixed-point divergence begins near alpha_c approximately 1/lambda_max(J), with the damped model remaining stable for alpha below 0.8/lambda_max. Measure whether free-energy descent and wall-profile error improve out of distribution.