Learning of Statistical Field Theories

Abstract

Recovering microscopic couplings directly from data provides a route to solving the inverse problem in statistical field theories, one that complements the traditional-often computationally intractable-forward approach of predicting observables from an action or Hamiltonian. Here, we propose an approach for the inverse problem that uniformly accommodates systems with discrete, continuous, and hybrid variables. We demonstrate accurate parameter recovery in several benchmark systems-including Wegner's Ising gauge theory, $φ^4$ theory, Schwinger and Sine-Gordon models, and mixed spin-gauge systems, and show how iterating the procedure under coarse-graining reconstructs full non-perturbative renormalization-group flows. This gives direct access to phase boundaries, fixed points, and emergent interactions without relying on perturbation theory. We also address a realistic setting where full gauge configurations may be unavailable, and reformulate learning algorithms for multiple field theories so that they are recovered directly using observables such as correlations from scattering data or quantum simulators. We anticipate that our methodology will find widespread use in practical learning of field theories in strongly coupled regimes where analytical tools might fail.

Figures (7)

• Figure 1: (a) Schematic of forward and inverse approaches to model validation. A traditional approach to validating a field theory would consist in computing an observable and comparing with an experiment, which is limited to tractable regimes of weak couplings. We put forward an alternative proposal, where model parameters are learned from data and compared to the postulated theory, an apporach which is not limited by the coupling strength. (b,c) Learning from correlations. Error plots for 1D Ising model and $\phi^4$ theory, learning from measured correlations for both large and small coupling values separated by order of magnitude and averaged over 20 repetitions at each step, demonstrating accurate and consistent parameter recovery for both discrete and continuous parameter models not only for weak coupling, but also in the case of strong coupling where perturbation theory fails.
• Figure 2: Learning renormalization group (RG) flows in discrete lattice models. (a) For the 1D Ising model, we depict a schematic of the block-spin procedure, and comparison between theoretical and learned RG trajectories for nearest-neighbor coupling $J$ and field $h$, and (b) For the 2D Wegner's Ising gauge theory, we show a schematic of the coarse-graining procedure on the gauge links, the evolution of the theoretical and estimated normalized gauge coupling $K$ and (c) Schematic for estimating 'String Tension' $\sigma$, with evolution of theoretical and estimated $\sigma$ under RG.
• Figure 3: Learning renormalization group (RG) flows in continuous models: $\phi^4$, Schwinger, and Sine–Gordon. (a) Phase structure and representative RG flows in 2D $\phi^4$ theory. (b) Schwinger model and its dual: schematics with RG flows for $\beta,m$ with characteristic stalling when length scale $a \sim m_{\gamma}^{-1}$ and for $\lambda,\mu^2$. (c) Sine–Gordon: flows in the $(\alpha,\beta)$ plane and the evolution of $\alpha,\beta$, $u$(coefficient of higher harmonic term, decaying at low energies (IR)), $Z$ (wavefunction renormalization, growing in IR).
• Figure 4: RG Flows and Complexity Plots for Mixed Model parameters $\beta$ and $K$ for one step. (a) RG Flow for $\beta$ and $K = 0.1$, with a non-trivial fixed point at $\beta$ close to $1.8$. (b) Complexity for $\beta$ versus sample size at $K = 0.1$. (c) Complexity for $K$ with exponential fit versus sample size.
• Figure 5: Learning from correlations only: (a) 1D Ising: learning the couplings $J$ and $h$ from moments alone, (b) 2D $\phi^4$: learning the parameters $\lambda$ and $m^2$ from correlations only. In both cases, the errors scale as $1/\sqrt{N}$ with the number of samples $N$, demonstrating efficient learning.