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Partial-Moment PINNs for Caldeira--Leggett Parameter Learning in Quantum Brownian Motion
Krishna Bhatia
TL;DR
Partial-moment observations make diffusion parameters in Caldeira–Leggett oscillator dynamics weakly identifiable. The paper develops physics-constrained moment-level PINNs and finds that constraints substantially improve dynamical rollout fidelity while retaining competitive parameter recovery and extending to slowly time-varying HPZ coefficients.
Problem
Partial moment observations leave the diffusion sector weakly identifiable, motivating methods for recovering Caldeira–Leggett parameters when momentum measurements are sparse or missing.
Method
A moment-level PINN predicts five moments, enforces CL/HPZ ODE residuals, and imposes positivity, fluctuation–dissipation structure, and sparse covariance anchors.
Results
Constrained PINNs reduce rollout RMSE to ≈0.11 versus ≈0.77 unconstrained while preserving positivity, recovering key parameters competitively and extending to time-varying HPZ coefficients.
Takeaways & Limitations
Adding one variance observable markedly improves conditioning, while sparse σpp anchors support stable recovery from only three observed moments.
Takeaways & Limitations
Uncertainty intervals come from a reduced bootstrap protocol that differs from the main setting, limiting their direct comparability.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION · II. PROBLEM FORMULATION
The paper frames high-temperature Caldeira–Leggett quantum Brownian motion as a moment-level linear stochastic system whose drift, diffusion, and fluctuation–dissipation structure govern the dynamics. It targets parameter learning from noisy partial moment observations while preserving physicality and benchmarking against exactly discretized classical estimators.
- I. INTRODUCTION: Quantum Brownian motion is modeled as a harmonic oscillator linearly coupled to an Ohmic bath, with a high-temperature Markovian Wigner description.The resulting phase-space state is (x, p), with mass m, frequency ω > 0, and damping γ ≥0.
- II. PROBLEM FORMULATION: The first and second moments obey a closed linear ODE whose drift contains γ and ω, while diffusion contains Dxx, Dxp, and Dpp.The formulation explicitly states that the first and second moments obey the model equations.
- II. PROBLEM FORMULATION: High-temperature CL assumptions enforce Dpp = 2mγkBT and Dxp ≈0, while a small Dxx > 0 supports complete positivity.The introduction also notes that Dxx ≥0 is added for complete positivity/Lindblad compatibility.
- I. INTRODUCTION: Realistic experiments expose only partial, noisy moments such as position mean and variance, making the diffusion sector weakly identifiable.The paper’s target is learning from incomplete moment traces rather than full phase-space observations.
- I. INTRODUCTION: The proposed PINN operates directly on moments, incorporates the same physics, and maintains positive covariance structure during parameter recovery.The stated model uses a moment-level PINN with physical constraints rather than estimating derivatives from raw traces.
- II. PROBLEM FORMULATION: Classical finite-difference and Kalman filtering/EM baselines require exact continuous-to-discrete dynamics for accurate estimation.The formulation uses sampling step ∆t for the exact discrete-time comparison model.
- II. PROBLEM FORMULATION: The baseline obtains Φ and Qd in one pass from the Van Loan block exponential.This provides the exact discrete-time model used for comparison.
III. METHOD: PARTIAL-MOMENT PINNS
The method represents five moments as smooth functions of normalized time using a Fourier-feature MLP and enforces covariance positive semidefiniteness through a Cholesky parameterization. Fourier features help localize ω, while the PSD constraint addresses long, low-frequency transients from planted ω = 1.0.
- Moment representation: The five moments are modeled as smooth functions of normalized time with a small tanh-activated MLP augmented by Fourier features.The Fourier features are included to help localize ω.
- Covariance constraint: The covariance is parameterized through a Cholesky factor so that Σ̂(t) = L(t)L(t)ᵀ ⪰ 0 for all t.This parameterization guarantees a positive semidefinite covariance.
- Covariance constraint: The PSD constraint is necessary because planted ω = 1.0 produces long, low-frequency transients on [0, 20].The transient behavior motivates enforcing covariance validity throughout the time interval.
A. Physics residual
The PINN forms scaled physics residuals at collocation times using autodifferentiated time derivatives. Its coefficients are either global for LTI dynamics or generated by a two-layer head for time-varying HPZ dynamics.
- Residual construction: Scaled residuals of equations (3b)–(3e) are formed at collocation times C using autodiff-derived ˙fϕ(t).The residual construction enforces the stated moment equations through automatic differentiation.
- Coefficient parameterization: The coefficient set (γ, ω, Dxx, Dpp, Dxp) is treated as global in the LTI setting.These five coefficients parameterize the time-independent model configuration.
- Coefficient parameterization: A small 2-layer coefficient head produces the coefficients in the time-varying Hu–Paz–Zhang setting.The same coefficient set is therefore represented dynamically for HPZ evolution.
B. Loss
The loss combines an observed-channel data term with scaled moment-equation residuals computed using automatic differentiation. The constrained formulation further uses weighted equations, sparse variance anchors, diffusion constraints, and selected priors or steady-state penalties.
- Loss: The loss includes an observed-channel data term and a physics term based on scaled residuals of the moment equations.Automatic differentiation obtains the predicted moment derivative ˙fϕ(t).
- Loss: Fixed per-equation weights are used, with the σxx equation upweighted in the constrained setting.Sparse σpp anchors are added to reduce diffusion non-identifiability under partial observability.
- Loss: The main constrained LTI setting enforces the high-T fluctuation-dissipation relation as a hard constraint with T = 1 and sets Dxp = 0.Under this setting, LFDT = 0; relaxing the tie introduces a soft penalty instead.
- Loss: Reported constrained LTI results use hard-FDT with Dxp = 0, strong σpp anchors, and a small Dxx prior.A weak end-of-window steady-state penalty can also be enabled.
C. Implementation details
The implementation normalizes time, encodes it with Fourier features, and uses a tanh MLP with a Cholesky covariance head to predict moments. LTI parameters are trained as global scalars, while HPZ runs use a time-dependent coefficient head and collocation residual scaling.
- Network and parameterization: Time is normalized to t′ = (t −t0)/(t1 −t0) and passed through Fourier features with kmax = 3.
- Network and parameterization: A 4-layer width-256 tanh MLP predicts (ˆµx, ˆµp) and Cholesky-head parameters constructing (ˆσxx, ˆσpp, ˆσxp).
- Coefficient learning: For LTI runs, (ω, γ, Dxx, Dpp, Dxp) are global trainable scalars, whereas HPZ-style runs use a two-layer head for time-varying coefficients.
- Residual evaluation: Residuals are evaluated on a uniformly sampled collocation set with |C| ≈3Ntrain and per-equation scaling.
IV. EXPERIMENTS
The experiments use two synthetic settings based on moment-ODE integrations, with partial noisy observations and a train/validation split. Sparse σpp anchors are also tested to remove diffusion degeneracy identified by a Fisher check.
- Two synthetic settings are evaluated, including an LTI Caldeira–Leggett case with planted parameters.
- The moment ODEs are integrated over t ∈ [0, 20] using N = 600 RK4 steps.
- Only {µx, σxx, σxp} are observed in both settings, with i.i.d. Gaussian readout noise σ = 0.01 and a 70%/30% train/validation split.
- 30 σpp anchors place at least one value every ≈0.7 time units on [0, 20], the smallest density removing diffusion degeneracy in the Fisher check.
A. Baselines
The baselines comprise a finite-difference estimator for (γ, ω) and diffusion terms, plus a Kalman–EM estimator using exact continuous-to-discrete covariance conversion. The latter provides a statistically efficient benchmark for evaluating PINN constraints.
- Finite-difference baseline: The finite-difference method estimates (γ, ω) from the mean equation and infers diffusion terms from restricted covariance equations.It uses ¨µx + γ ˙µx + ω2µx = 0 for parameter estimation.
- Kalman–EM baseline: The Kalman–EM estimator operates on the latent (x, p) state and converts (A, Qc) to (Φ, Qd) using the Van Loan block exponential.This supplies exact discretization for the state-transition and process-noise covariance matrices.
- Kalman–EM baseline: The Kalman–EM procedure maps discrete estimates back to continuous-time parameters via log(Φ)/∆t and Qd ≈Qc∆t.It is used as a statistically efficient baseline for judging the benefit of PINN constraints.
B. Metrics · C. Additional diagnostics · V. RESULTS
The paper evaluates parameter recovery and predictive fidelity with network and rollout RMSE, while monitoring covariance positivity. It complements the main comparison with Fisher-conditioning and bootstrap uncertainty diagnostics, using exact Van Loan transitions for classical baselines.
- B. Metrics: Predictive quality is measured by network RMSE between fϕ(t) and ground truth, and rollout RMSE from integrating learned coefficients in the moment ODE.
- B. Metrics: The evaluation monitors λmin(Σ̂(t)) to assess whether the predicted covariance remains PSD.
- C. Additional diagnostics: Additional diagnostics examine Fisher-conditioning across different observed subsets to assess partial-observability effects.
- C. Additional diagnostics: Bootstrap uncertainty is reported for the constrained PINN as a second diagnostic relevant to partial observability.
- V. RESULTS: The results cover parameter recovery, predictive fidelity, identifiability diagnostics, and uncertainty quantification.
- V. RESULTS: All classical baselines use exact Van Loan transitions.
A. Parameter estimates (LTI) · B. Predictive fidelity (RMSE) · C. Identifiability diagnostics
Across LTI experiments, constrained PINNs are evaluated against planted parameters and established baselines, with improved dynamical rollouts despite a small direct-RMSE cost. Fisher analyses show that variance observations substantially improve parameter identifiability over mean-only measurements.
- A. Parameter estimates (LTI): Table II compares planted parameters with finite differences, Kalman–EM, and unconstrained and constrained PINNs.PINN summaries average three random seeds.
- B. Predictive fidelity (RMSE): The constrained variant pays a small price in direct network RMSE but produces far more accurate ODE rollouts.Prediction and rollout errors are reported as mean±standard deviation in Table III.
- A. Parameter estimates (LTI): The planted LTI parameters are (γ, ω, Dpp, Dxx) = (0.25, 1.0, 0.50, 10−3).The table reports PINN values averaged over three seeds.
- B. Predictive fidelity (RMSE): For µx, µp, and σxx, constrained PINN rollouts better respect the dynamics than direct predictions, especially for σxx.The figure compares truth, direct PINN predictions, and learned-coefficient ODE rollouts.
- C. Identifiability diagnostics: Mean-only observations are extremely ill-conditioned in finite-difference Fisher analyses.Adding even one variance channel drastically improves conditioning.
- C. Identifiability diagnostics: Adding even one variance channel drastically improves conditioning for parameter recovery.This improvement is reported in Table IV’s finite-difference Fisher analysis.
D. Uncertainty quantification · E. HPZ (time-varying) summary · VI. CONCLUSION
The paper reports cautiously interpreted bootstrap uncertainty, good coverage for γ and Dpp but weaker coverage for ω and Dxx, and stable HPZ learning for slowly time-varying coefficients. Its conclusion emphasizes constrained Partial-Moment PINNs, which preserve positivity and improve rollout fidelity from partial moment observations.
- D. Uncertainty quantification: Bootstrap intervals from B=10 resamples and a shortened training schedule should not be interpreted as confidence intervals around Table II point estimates.The reduced protocol differs from the main setting used in Table II.
- D. Uncertainty quantification: The Fisher analysis compares condition numbers across observed subsets, with lower values indicating better conditioning.
- D. Uncertainty quantification: Coverage is good for γ and Dpp but not for ω and Dxx, consistent with partial channels and the positivity nudge on Dxx.
- E. HPZ (time-varying) summary: The HPZ–hardFDT coefficient head learns smooth γ(t) and Dxx(t) curves while constructing Dpp(t) = 2γ(t)T.
- E. HPZ (time-varying) summary: The HPZ–hardFDT physics residual is ≈3.6 × 10^-3.
- E. HPZ (time-varying) summary: HPZ rollout RMSE is within 5% of the LTI constrained case, indicating stability for slowly time-varying CL coefficients.
- VI. CONCLUSION: The proposed method identifies CL parameters from partially observed moment trajectories while enforcing PSD, hard high-T fluctuation–dissipation structure, and sparse σpp anchors.
- VI. CONCLUSION: Compared with the unconstrained PINN, the constrained model improves rollout RMSE from ≈0.77 to ≈0.11 while preserving positivity with only three of five moments observed.