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This paper introduces a stochastic renormalization group (RG) framework combined with path integrals to learn a hierarchy of coarse-grained spatiotemporal dynamics. The RG diffusion process applies scale-dependent Laplacian damping and Gaussian noise, while time evolution at each scale is modeled via an Onsager-Machlup action with an RG-determined predictor and noise amplitude. The predictor is optimized by minimizing KL divergence between RG and path-integral distributions, yielding a unified formulation for simulation, unconditional generation, and super-resolution.
A novel stochastic RG framework learns a hierarchy of multiscale dynamics by unifying predictor and score functions, enabling simulation, generation, and super-resolution from a single learned model.
This study combines a stochastic renormalization group (RG) in space with a path-integral description of the time evolution, giving a spatial hierarchy of coarse-grained dynamics together with the distribution over spatiotemporal paths. The RG is defined as a diffusion process that applies scale-dependent Laplacian damping together with additive Gaussian noise. The time evolution at each spatial scale is formulated as an Onsager--Machlup action, with a drift (i.e., the predictor) and white noise whose amplitude is fixed by the RG. The predictor of these dynamics is optimized by minimizing the Kullback--Leibler divergence between the path distributions from the RG and from the path-integral description. The optimal predictor contains the score function that connects the spatial scales, so the predictor and the score are two aspects of the same multiscale path formulation. The formulation unifies simulation using the predictor, unconditional generation using the score, and super-resolution using both. The predictor has no closed form, so it is computed by a neural network in two realizations that differ only in how the score is computed. The first realization obtains the score by automatic differentiation of the path distribution, remaining faithful to that formulation. The second obtains the score as an additional network output trained by denoising, at a lower computational cost. These two realizations are validated through numerical experiments on two representative multiscale systems, the Kolmogorov flow and the two-timescale Lorenz-96 model.