Rediscovery of Numerical Lüscher's Formula from the Neural Network

TL;DR

This work addresses how to extract a model-independent relationship between continuum scattering data and finite-volume spectra by training a compact neural network to map phase shifts $\delta(E)$ and lattice size $L$ to the finite-volume energies $E(L)$. Using Hamiltonian Effective Field Theory data for single-channel elastic $S$-wave $\pi\pi$ scattering and three potential shapes, the authors show the network can reproduce the numerical form of Lüscher's formula with high precision and robust generalization. The study demonstrates that a data-driven approach can uncover deep physical principles, offering a practical route to translate lattice spectra into continuum observables while mitigating model dependence. This has potential implications for lattice QCD analyses and the discovery of model-independent relations from complex data.

Abstract

We present that by predicting the spectrum in discrete space from the phase shift in continuous space, the neural network can remarkably reproduce the numerical Lüscher's formula to a high precision. The model-independent property of the Lüscher's formula is naturally realized by the generalizability of the neural network. This exhibits the great potential of the neural network to extract model-independent relation between model-dependent quantities, and this data-driven approach could greatly facilitate the discovery of the physical principles underneath the intricate data.

Figures (7)

• Figure 1: The workflow of this work.
• Figure 2: The structure of our neural network. Green round rectangles with integer $n$ represent the linear layer with size $n$, which consists of all the learning parameters. Orange circles denote the input and output nodes and blue circles are layers with operations marked in the middle. The yellow thick arrow marks the "SoftPlus" activation function and the right brace is a conjunction of the corresponding layers.
• Figure 3: The histogram of $\Delta(E)\equiv E_{\mathrm{model}} - E_{\mathrm{NN}}$ at $L=10$ fm, where $E_{\mathrm{model}}, E_{\mathrm{NN}}$ represent the predictions from the neural network and the model, respectively. The neural network is trained on the data from model A and C, and the data from model B serves as test set.
• Figure 4: Comparison of the Lüscher's formula (red), predictions from the neural network (black) and models (blue), where lattice size is $10$ fm.
• Figure 5: Comparison of model C (blue points) with the Lüscher's formula in different volume sizes.