PI-LNO: Physics-Informed Latent Neural Operator for Coupled HJB-KFE Equilibria

We introduce PI-LNO, a physics-informed neural operator for jointly learning structured probabilistic and functional outputs---a stationary density $\mu(a,z)$, value function $V(a,z)$, and endogenous price $r^*$---that must simultaneously satisfy adjoint PDE constraints (HJB and KFE) and a market-clearing closure. Constrained output heads enforce measure validity and value-policy consistency by construction; a three-stage curriculum gates physics residuals progressively to prevent destabilizing gradient feedback during density learning. On 7,500 solver-computed equilibria across a seven-dimensional parameter space, PI-LNO achieves 0.45\% value-function MAE and 0.20\% Gini MAE (Stage~3, five-seed mean) with $7{,}590\times$ batched speedup enabling amortized simulation-based inference; FiLM conditioning contributes $\sim$$2.4\times$ on value error, and the surrogate's speed permits identification diagnostics that expose weak global identifiability along $(\sigma, \bar{z})$.