Operator Learning in Lattice QCD: Spectral Reconstruction

Technical Summary: Operator Learning in Lattice QCD: Spectral Reconstruction

Problem Statement
Spectral reconstruction in Lattice Quantum Chromodynamics (QCD) involves extracting spectral densities from Euclidean correlation functions computed via Monte Carlo simulations. This task is mathematically equivalent to the numerical inversion of a Laplace transform, a problem that is notoriously ill-posed due to statistical noise in the correlators and the limited amount of available data points. While methods like the Hansen-Lupo-Tantalo (HLT) algorithm provide model-independent solutions with controlled uncertainties, they rely on linear combinations of correlator values. This linear approach often leads to a trade-off where reducing systematic uncertainties requires accepting large statistical uncertainties, particularly for sharply peaked smearing kernels. Previous machine learning approaches (e.g., Ref. [25]) treated spectral reconstruction as a vector-to-vector mapping, requiring retraining for every change in input time grids or output parameters, rendering them computationally impractical for broad phenomenological applications.

Methodology
This work proposes a novel supervised learning strategy based on Operator Learning, specifically utilizing Deep Operator Networks (DeepONets). Instead of mapping fixed vectors to fixed vectors, the proposed framework trains neural networks to approximate the operator $\hat{O}$ that maps the space of Euclidean correlator functions to the space of smeared spectral functions.

• Architecture: The method employs a DeepONet structure consisting of two independent neural networks:
  • A Branch Net that encodes the input Euclidean correlator (represented as a vector on a fixed grid of Euclidean times) into a latent representation.
  • A Trunk Net that encodes the domain of the output (energy $E$ and smearing parameter $\sigma$) into a corresponding latent representation.
  • The final prediction is the dot product of these two latent vectors, allowing for continuous evaluation of the spectral density at any $(E, \sigma)$ without retraining.
• Training Data Generation: The networks are trained on mock data generated from a Gaussian Process (GP). The GP is constructed with a mean function corresponding to the known two-particle contribution of the $O(3)$ non-linear $\sigma$ model and a covariance kernel that allows for smooth variations above the four-particle threshold. This incorporates prior physical knowledge while maintaining a broad space of admissible functions.
• Uncertainty Quantification: To address the systematic uncertainty inherent in neural network approximations, the authors utilize an ensemble of 15 independently trained networks with varying architectures (depth, width) and training set sizes.
  • Systematic Uncertainty: Estimated via the weighted spread of predictions from the ensemble, where weights are assigned based on the minimum validation loss of each network.
  • Statistical Uncertainty: Estimated via bootstrap resampling of the input correlators.
  • Global Bias: Corrected by evaluating the mean difference between predictions and true values on a separate set of mock data.
• Noise Handling: The training sets include noise injected into the mock correlators to match the noise-to-signal ratios of actual lattice ensembles (specifically ensembles A1–A4, B1, and B2 from Ref. [4]).

Key Contributions

1. First Application of DeepONets in Lattice QCD: This work introduces the DeepONet architecture to the field, demonstrating its ability to learn the operator mapping Euclidean correlators to smeared spectral densities.
2. Overcoming Retraining Limitations: By treating the problem as operator learning, the method allows for predictions on correlators defined on different time grids or for different output parameters without retraining the network, a significant improvement over previous vector-to-vector ML approaches.
3. Robust Uncertainty Estimation: The paper establishes a rigorous procedure for estimating systematic uncertainties using neural network ensembles, validated through closure tests on unseen mock data.
4. Benchmark Validation: The strategy is applied to the 1+1-dimensional $O(3)$ non-linear $\sigma$ model, a benchmark system with a known analytic spectral density. The method successfully reconstructs the inclusive rate above the four-particle threshold up to high energies.

Results
The proposed strategy was validated on both mock data and actual lattice correlators from Ref. [4].

• Closure Tests: On unseen mock data, the method demonstrated that the estimated total uncertainty (combining statistical and systematic errors) is conservative, with the "pull" distribution (deviation from truth normalized by error) consistent with Gaussian expectations.
• Lattice Application: The method was applied to reconstruct the spectral density of the $O(3)$ model. The final result, obtained after performing infinite-volume, continuum, and $\sigma \to 0$ limits, is in excellent agreement with the exact analytic spectral density.
• Comparison with HLT: Compared to the state-of-the-art HLT method (Ref. [4]), the DeepONet-based strategy exhibits a significant reduction in total uncertainty. The final determination achieves a relative uncertainty of approximately 2% for energies $E < 20m$, rising to about 9% at higher energies. The error budget shows that statistical, reconstruction, finite-size, and continuum uncertainties contribute at comparable levels, with no single source dominating.

Significance and Claims
The authors claim that this work demonstrates the potential of DeepONets to provide a more powerful approach for extracting spectral information from lattice QCD simulations compared to traditional linear regularization methods. The method successfully recovers the known analytic spectral density with higher precision than the HLT algorithm.

The paper maintains a modest tone regarding its broader applicability. While the results are promising, the authors explicitly state that the extent to which the observed uncertainty reduction persists in the absence of the specific prior physical knowledge (the GP mean function based on the $O(3)$ model) "remains to be quantified." They note that the strategy is naturally extendable to more phenomenologically relevant observables and other operations in lattice field theory, but the current validation is strictly limited to the $O(3)$ model benchmark. The work does not claim to solve the spectral reconstruction problem for arbitrary, unknown physical systems without any prior constraints, but rather establishes a proof-of-concept for a new, operator-learning-based paradigm that outperforms existing benchmarks in a controlled setting.