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 Dxp≈0, and fluctuation--dissipation ties between Dpp and γ. On synthetic CL data with channels μx,σxx,σxp, the constrained variant recovers (ω,γ) accurately, stabilizes Dpp, 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 σpp ``anchors'' restore conditioning. We also show that the same PINN can learn time-varying HPZ coefficients.