Predictor-Driven Diffusion for Spatiotemporal Generation

Abstract

Multiscale spatial structure complicates temporal prediction because small-scale spatial fluctuations influence large-scale evolution, yet resolving all scales is often intractable. Standard diffusion models do not address this problem effectively since they apply uniform decay across all Fourier modes. We propose Predictor-Driven Diffusion, a framework that combines renormalization-group-based spatial coarse-graining with a path-integral formulation of temporal dynamics. The forward process applies scale-dependent Laplacian damping together with additive noise, producing a hierarchy of coarse-grained fields indexed by diffusion scale $λ$. Training minimizes the Kullback-Leibler divergence between data-induced and predictor-induced path densities, leading to a simple regression loss on temporal derivatives. The resulting predictor captures how eliminated small-scale components statistically influence large-scale evolution. A key insight is that the same predictor provides a path score for reverse-$λ$ sampling, unifying simulation, unconditional generation, and super-resolution in a single framework. Our unified approach is validated through experiments on two multiscale turbulent systems.

Figures (17)

• Figure 1: Schematic of the proposed framework. The horizontal axis represents the diffusion scale $\lambda$, which controls the degree of coarse-graining; the vertical axis represents physical time $t$. Blue box: forward simulation in $t$ emulates dynamics at any fixed $\lambda$; red box: reverse-$\lambda$ integration of a spatiotemporal sample enables generation from noise and super-resolution from a coarse-grained input. A single predictor, trained by minimizing KL divergence between path densities (bottom box), supports all three tasks (i.e., simulation, generation, and super-resolution).
• Figure 2: Simulation results for the Lorenz-96 model at fine resolution ($\lambda = 0$, top) and coarse-grained resolution ($\lambda=0.2$, bottom). Left: slow variable $X$; right: fast variable $Y$. Each panel shows spatiotemporal evolution on the left and time-averaged spatial power spectral density (PSD) on the right. The surrogate model accurately reproduces both the spatiotemporal patterns and spectral statistics of physics-based simulations. All quantities are non-dimensionalized.
• Figure 3: Simulation results for the Kolmogorov flow at fine resolution ($\lambda = 0$, top) and coarse-grained resolution ($\lambda=0.2$, bottom). Left: vorticity field at the sixth time step of the surrogate simulation; right: time-averaged spatial power spectral density (PSD). Coarse-graining removes small-scale structure while preserving large-scale dynamics. All quantities are non-dimensionalized.
• Figure 4: Super-resolution results for the Lorenz-96 model (top) and Kolmogorov flow (bottom), shown at the sixth time step. Left: super-resolved output at $\lambda = 0$; middle: low-resolution input from surrogate simulation at $\lambda=0.2$; right: time-averaged spatial power spectral densities (PSD; computed from 100 samples) comparing low-resolution input, super-resolved output, and physics-based simulation. The reverse-$\lambda$ integration restores small-scale structure absent from the low-resolution input. All quantities are non-dimensionalized.
• Figure 5: Simulation results at $\lambda = 0$ for the Lorenz-96 model, comparing our surrogate model with physics-based simulation and the DDPM baseline. Left: slow variable $X$; right: fast variable $Y$. Top: each panel shows physics-based simulation, surrogate prediction, and DDPM baseline; bottom: corresponding spatiotemporal power spectral densities (PSDs; computed from 100 samples). Both learned models accurately reproduce the spatiotemporal patterns and spectral statistics. All quantities are non-dimensionalized.