Finding low-dimensional interpretable models of complex physical fields such
as turbulence remains an open question, 80 years after the pioneer work of
Kolmogorov. Estimating high-dimensional probability distributions from data
samples suffers from an optimization and an approximation curse of
dimensionality. It may be avoided by following a hierarchic probability flow
from coarse to fine scales. This inverse renormalization group is defined by
conditional probabilities across scales, renormalized in a wavelet basis. For a
$\varphi^4$ scalar potential, sampling these hierarchic models avoids the
critical slowing down at the phase transition. An outstanding issue is to also
approximate non-Gaussian fields having long-range interactions in space and
across scales. We introduce low-dimensional models with robust multiscale
approximations of high order polynomial energies. They are calculated with a
second wavelet transform, which defines interactions over two hierarchies of
scales. We estimate and sample these wavelet scattering models to generate 2D
vorticity fields of turbulence, and images of dark matter densities.

利用二重小波变换计算的低维度模型，通过多尺度逼近高阶多项式能量，估计和采样小波散射模型，产生湍流的二维涡量场和暗物质密度图像。