PAC Studio Machine Learning: Human-in-the-Loop Analysis of TDPAC Spectra

Technical Summary: PAC Studio ML

Problem Statement
Time-differential perturbed angular correlation (TDPAC or PAC) spectroscopy is a powerful local probe technique for determining hyperfine parameters (electric-field gradients, magnetic fields, site fractions) in condensed matter. However, extracting these physical parameters from experimental spectra constitutes an ill-conditioned inverse problem. Measured $R(t)$ curves are compressed projections of microscopic configurations where different combinations of site counts, interaction types, angular parameters, and damping terms can produce visually similar spectral shapes. Consequently, conventional nonlinear least-squares fitting is highly sensitive to initialization, site-model hypotheses, and parameter bounds. Manual exploration of these variables is time-consuming, difficult to reproduce, and prone to converging on local minima that lack physical uniqueness. While machine learning (ML) offers potential for inverse analysis, existing approaches often lack transparency, fail to integrate with conventional fitting workflows, or treat ML as a "black box" that replaces expert judgment rather than supporting it.

Methodology
The paper introduces PAC Studio ML, a human-in-the-loop Python desktop environment designed to support the pre-fitting and model-screening stages of PAC analysis. The software architecture is modular, integrating a physics-based forward solver, machine-learning predictors, and standard least-squares fitting tools within a single graphical user interface (GUI).

• Forward Physics Engine: The core physics layer utilizes a Hamiltonian-matrix formalism to calculate combined electric and magnetic hyperfine interactions. It generates synthetic spectra by diagonalizing the Hamiltonian to obtain transition frequencies and constructing perturbation functions $G_{22}(t)$ using Wigner 3j coefficients. These are damped (Gaussian or Lorentzian) and weighted by site fractions to produce the observable signal $R(t) = \text{add\_const} + A_{22} G_{22}(t)$.
• Synthetic Data Generation: Instead of relying on fixed, universal training sets, the software allows users to define physical validity domains (ranges for $v_Q$, $v_M$, $\eta$, damping, noise, etc.) to generate custom synthetic databanks. Separate libraries are created for one-, two-, and three-site models, with data split into training (80%), validation (10%), and held-out test (10%) sets.
• Feature Extraction: Raw $R(t)$ curves are converted into numerical feature vectors comprising four families: time-domain descriptors (amplitude, moments, slopes), frequency-domain descriptors (FFT peaks, bandwidth), shape descriptors (resampled/normalized profiles), and physics-informed descriptors (angular-frequency proxies, interaction indicators).
• Machine Learning Models: The system employs supervised learning to map feature vectors to target parameters. It utilizes tree-based regressors (Random Forest, Extra Trees, HistGradientBoosting) from scikit-learn, with a k-nearest-neighbor fallback for environments lacking these libraries. Angular parameters are handled via sine/cosine decomposition to avoid discontinuity.
• Prediction and Refinement Workflow:
  1. Direct Prediction: The trained model proposes candidate parameters based on the input spectrum's feature vector.
  2. Auto Sites Assistant: This tool scans synthetic databanks for one-, two-, and three-site models, ranking candidates using RMS residuals, BIC-style complexity scores, and feature distances.
  3. ML-Seeded Refinement: Predicted parameters serve as initialization for conventional nonlinear least-squares fitting.
  4. Safeguards: Post-prediction corrections include FFT peak alignment, robust amplitude/baseline calibration, and residual-based reranking of candidates against the experimental curve.
• Reproducibility Infrastructure: The software includes tools for benchmarking initialization strategies, ablation testing of correction components, and empirical uncertainty calibration. It generates "Model Cards" documenting training ranges, feature definitions, and known failure modes to ensure auditability.

Key Contributions

• Integrated Human-in-the-Loop Environment: PAC Studio ML bridges the gap between synthetic modeling, data-driven inverse prediction, and expert-led refinement, ensuring that ML suggestions remain inspectable and controllable.
• Physics-Informed Feature Engineering: The system translates PAC spectra into descriptors that capture both general signal shape and specific physical signatures (e.g., distinguishing pure electric from combined interactions).
• Transparent Model Selection: The "Auto Sites" assistant provides a systematic method for comparing site-count hypotheses, penalizing unnecessary complexity while exposing alternative plausible models.
• Reproducibility and Diagnostics: The inclusion of model cards, ablation benchmarks, and uncertainty calibration tools addresses the "black box" nature of typical ML applications in science, allowing users to assess the reliability and scope of predictions.

Results
The paper presents "proof-of-operation" results using synthetic held-out data and illustrative experimental examples from BiFeO$_3$ (BFO):

• Synthetic Recovery: On held-out synthetic test sets, global parameters (amplitude $A_{22}$ and baseline) were recovered with high accuracy. Magnetic frequencies ($v_M$) showed moderate recoverability. However, site-specific quadrupole frequencies ($v_Q$), asymmetry parameters ($\eta$), angular variables, and damping terms exhibited higher errors, particularly in multi-site scenarios. This confirms the non-identifiability of certain parameters in ill-conditioned PAC problems.
• Initialization Efficiency: In a BFO nanoparticle case study, ML-seeded fitting converged faster (285 function evaluations vs. 303 for manual) and achieved a lower reduced chi-square compared to manual initialization. It significantly outperformed random starts, which had a lower success rate and higher computational cost.
• Model Ambiguity: In a bulk BFO case, the "Auto Sites" tool proposed a two-site combined-interaction model, while a literature-informed one-site pure-electric model provided a similarly good fit. This demonstrates that multiple model families can reproduce experimental data, reinforcing the necessity of expert judgment in selecting the physically meaningful solution.

Significance and Claims
The authors position PAC Studio ML strictly as a supporting tool for expert PAC analysis, not a replacement for it. The paper explicitly claims that the software:

• Accelerates the exploration of parameter space and site-count hypotheses.
• Improves the efficiency and stability of nonlinear least-squares initialization.
• Enhances the reproducibility and diagnostic transparency of PAC analysis workflows.
• Makes the limitations of ML predictions (via uncertainty estimates and model cards) visible to the user.

The paper modestly asserts that it does not claim universal physical accuracy, calibrated uncertainty for all experimental spectra, or the resolution of all PAC ambiguities. The synthetic tests and BFO examples serve to demonstrate the software's operational capability and workflow integration. The authors state that comprehensive material-specific conclusions, full uncertainty calibration, and independent experimental validation are reserved for a subsequent physics-results paper. The primary contribution of this work is the definition of a reproducible, documented, and human-centered software platform for physics-informed inverse analysis.