Partial-Moment PINNs for Caldeira--Leggett Parameter Learning in Quantum Brownian Motion

We study parameter recovery in the Caldeira--Leggett (quantum Brownian) oscillator from partial moment traces. Our model is a moment-level PINN that predicts the five first/second moments and enforces the linear CL/HPZ ODEs by automatic differentiation. Physical structure is imposed through a PSD (Cholesky) covariance head, high-temperature CL assumptions with D_xpapprox0, and fluctuation--dissipation ties between D_pp and γ. On synthetic CL data with channels {μ_x,σ_xx,σ_xp}, the constrained variant recovers (ω,γ) accurately, stabilizes D_pp, and achieves low rollout error compared to finite differences and Kalman--EM (expectation--maximization) with exact Van Loan discretization. Fisher-style checks confirm that diffusion needs at least one variance observable, and sparse σ_pp "anchors" restore conditioning. We also show that the same PINN can learn time-varying HPZ coefficients.