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 $D_{xp}\approx0$, 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.

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