用物理约束的神经网络从部分观测数据中精准恢复量子布朗运动参数。
Partial-Moment PINNs for Caldeira--Leggett Parameter Learning in Quantum Brownian Motion
- 基于前两阶矩构建带物理约束的PINN，通过自动微分满足动力学方程。
- 在合成数据上准确恢复频率与耗散系数，误差低于有限差分和卡尔曼-EM方法。
- 适用于缺乏完整协方差信息的场景，尤其适合参数随时间变化的情况。
我们研究从部分矩迹线中恢复Caldeira--Leggett（量子布朗）振子的参数。模型为基于矩的PINN，可预测前五个一阶/二阶矩，并通过自动微分强制满足线性的CL/HPZ常微分方程。通过正定（Cholesky）协方差头、高温下$D_{xp}≈0$假设，以及$D_{pp}$与$γ$间的涨落-耗散关系施加物理结构。在含通道${μ_x,σ_{xx},σ_{xp}}$的合成数据上，该方法准确恢复$(ω,γ)$，稳定$D_{pp}$，且滚动误差显著低于有限差分与基于精确Van Loan离散化的卡尔曼-EM方法。费舍尔检验表明扩散项至少需一个方差观测，而稀疏$σ_{pp}$作为“锚点”可恢复条件性。此外，同一PINN亦可学习时变的HPZ系数。
原文摘要 · Abstract (English)
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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