Lee-Yang Phase-Diagram Loss for Neural Potentials

Integrate this as an auxiliary loss for a neural interatomic potential, neural Hamiltonian, or differentiable energy-based model. For each training state q_j = (T_j,p_j), run short equilibrium simulations using the current U_theta, collect configurations x_n and the extensive observable M_n associated with the field being complexified. For a liquid-gas transition, use particle number or volume; for a magnetic transition, use magnetization; for a structural transition, use an extensive bias variable that distinguishes the phases. Pseudocode: (1) sample x_n from the current model at q_j; (2) for each imaginary-field value h_i,k compute phi_hat_jk = (1/N_s) sum_n exp(i beta_j h_i,k M_n); (3) compute the log-modulus loss against target curves or target minimum locations; (4) backpropagate through differentiable energy evaluations or use a reweighting estimator with the simulation samples treated as fixed; (5) update theta and periodically resample because the distribution changes. The paper supplies the complex partition-function objective and the interpretation of zeros; sampling variance, target curves, and finite-size corrections must be estimated empirically. The first cheap experiment is a two-dimensional Lennard-Jones fluid with a small neural correction to the analytic potential, system sizes N = 64, 128, and 256, coexistence temperatures from reference simulation as targets, and ordinary energy/force fitting as baseline. The quantitative prediction is that the learned model's minimum h_i*(N) decreases approximately as N^(-1) at the target transition but remains bounded away from zero at off-transition temperatures. Accept the mechanism only if the inferred transition agrees with the reference within 20 percent and the fitted finite-size exponent is reasonably close to 1 for the first-order case. Held-out enthalpy and heat capacity should also be measured as secondary validation.