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 $D_{xp}\approx 0$, and fluctuation–dissipation ties between $D_{pp}$ and $\gamma$. On synthetic CL data with channels ${\mu }_x,{\sigma }_{xx},{\sigma }_{xp}$, the constrained variant recovers $(\omega ,\gamma )$ 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 ${\sigma }_{pp}$ ``anchors’’ restore conditioning. We also show that the same PINN can learn time-varying HPZ coefficients.