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 $\vvarphi^4$ scalar potential, sampling these hierarchic models avoids the critical slowing down at the phase transition. In a well chosen wavelet basis, conditional probabilities can be captured with low dimensional parametric models, because interactions between wavelet coefficients are local in space and scales. An outstanding issue is also to approximate non-Gaussian fields having long-range interactions in space and across scales. We introduce low-dimensional models of wavelet conditional probabilities with the scattering covariance. It is 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.

翻译：在科尔莫戈罗夫开创性工作80年后，寻找复杂物理场（如湍流）的低维可解释模型仍然是一个悬而未决的问题。从数据样本中估计高维概率分布受到优化维数灾难和近似维数灾难的困扰。通过遵循从粗尺度到细尺度的分层概率流，可以避免这一问题。这种逆重整化群由跨尺度的条件概率定义，并在小波基中重整化。对于$\vvarphi^4$标量势，对这些分层模型进行采样可避免相变时的临界减速现象。在适当选择的小波基中，由于小波系数间的相互作用在空间和尺度上具有局部性，条件概率可以用低维参数模型捕捉。另一个关键问题是如何近似在空间和跨尺度上具有长程相互作用的非高斯场。我们引入具有散射协方差的小波条件概率低维模型。该模型通过二次小波变换计算，定义了跨越两个尺度层次结构的相互作用。我们估计并采样这些小波散射模型，以生成二维湍流涡量场和暗物质密度图像。