Generative sampling with physics-informed kernels

TL;DR

This work introduces physics-informed kernels (PIKs) as a novel generative architecture for Monte Carlo sampling in lattice field theories. By recasting layerwise propagation as independent, analytically solvable linear PDEs for kernels via physics-informed renormalisation group (PIRG) flows, it transforms the global sampling task into a sequence of tractable local problems, mitigating out-of-domain failures. The approach supports independent kernel learning, systematic out-of-domain corrections, and parameter-conditional optimisation, demonstrated in zero-dimensional and high-dimensional $\phi^4$ theories, with notable speedups over traditional methods. The framework offers a scalable path to incorporating richer physics (e.g., gauge/fermions) and provides a principled route for error control and optimization in generative sampling for high-dimensional distributions.

Abstract

We construct a generative network for Monte-Carlo sampling in lattice field theories and beyond, for which the learning of layerwise propagation is done and optimised independently on each layer. The architecture uses physics-informed renormalisation group flows that provide access to the layerwise propagation step from one layer to the next in terms of a simple first order partial differential equation for the respective renormalisation group kernel through a given layer. Thus, it transforms the generative task into that of solving once the set of independent and linear differential equations for the kernels of the transformation. As these equations are analytically known, the kernels can be refined iteratively. This allows us to structurally tackle out-of-domain problems generally encountered in generative models and opens the path to further optimisation. We illustrate the practical feasibility of the architecture within simulations in scalar field theories.

Figures (8)

• Figure 1: Illustration of the construction principle of PIKs for degrees of freedom $\phi$ that are propagated through the layers $L_i$ using independent kernels $\dot{\phi}_i$.
• Figure 2: Training of most generative models from a distribution characterised by $F_{t=0}$ to one characterised by $F_{t=1}$. The path $F_t$ is not fully determined and accessible. Optimisable parameters $\theta_t$ at different times are entangled, which is indicates by the red entanglement line between the parameters $\theta_{t_1}$ and $\theta_{t_2}$.
• Figure 3: Training of PIKs from a distribution characterised by $F_{t=0}$ to one characterised by $F_{t=1}$. The path $F_t$ is fully determined and accessible. Optimisable parameters $\theta_t$ at different times are not entangled, which is indicated by the broken dashed red lines.
• Figure 4: Comparison of the true (orange) and the learned (black) kernel $\dot{\phi}_{t=1}(\phi)$ for the zero-dimensional $\phi^4$-theory.
• Figure 5: Comparison of the targeted (orange, HMC) and modelled (black, PIK) zero-dimensional $\phi^4$ distribution.