Neural-network parametrization of fundamental measure theory: From hard spheres to Lennard-Jones fluids

We introduce a neural-network framework for parametrizing fundamental measure theory (FMT) applied to hard-sphere and Lennard-Jones fluids. Lutsko's extended scalar FMT expresses the excess free-energy density through two parameters, A and B, which control the relative weight of tensor invariants in the third-order contribution Φ_{3}; different choices recover established functionals (Rosenfeld, White Bear, and the recent optimized values of Gül et al.). We treat A and B as learnable functions of the packing fraction η=(π/6)ρσ^{3}, predicted by a small neural network and optimized end to end through a JAX-based density functional solver. Four training strategies are examined, namely matching the Carnahan-Starling equation of state, minimizing chemical-potential and compressibility deviations, optimizing wall contact densities, and a combined multiobjective loss. All four converge to parameters near the Percus-Yevick line defined by the constraint combination C=8A+2B-9≈0, with subpercent agreement with Carnahan-Starling thermodynamics. Extension to Lennard-Jones fluids via Weeks-Chandler-Andersen perturbation theory, using the temperature-dependent Barker-Henderson effective diameter to define an effective packing fraction, yields vapor-liquid coexistence with a critical temperature T_{c}^{*}=k_{B}T_{c}/ε≈1.28, consistent with mean-field DFT benchmarks. The learned parameters remain close to the fixed Lutsko baseline (A=1, B=0) throughout, confirming that bulk phase equilibria alone cannot drive nontrivial density dependence. A three-phase training strategy-bulk equation-of-state fitting, test-particle sum-rule optimization, and wall-contact density fine tuning via numerical gradients-breaks this degeneracy, yielding contact densities within 1-2% of molecular dynamics data across all benchmark packing fractions and a 5.5-fold reduction in final wall-contact loss relative to direct wall fine tuning, at the cost of 5-8% bulk thermodynamic errors that quantify a genuine bulk-interface tradeoff. These results provide a starting point for adaptive density functionals trained on inhomogeneous simulation data.