Learning Invariant Measures with Physics Supervision

Many fluid and plasma applications require samples of statistically stationary turbulence to estimate mean observables, fluctuations, and spatial correlations. However, obtaining these samples typically involves long-time evolution, either during sampling or when constructing training datasets for generative models. We introduce the Invariant Measure Generator (IMG), which learns a direct sampler of long-time stationary statistics using supervision from short physical evolutions. Its self-consistent conditional flow-matching objective uses evolved model samples as targets, providing tractable supervision for high-dimensional fields without requiring stationary training samples. Our analysis connects the self-consistent objective to distributional invariance. Experiments on Lorenz-63, the Kuramoto-Sivashinsky equation, and the Hasegawa-Wakatani (HW) system demonstrate accurate stationary statistics, including spectra and spatial increments in two-dimensional turbulence. In limited-data settings, the same objective improves data-trained samplers. Using an existing frozen neural operator to construct its targets improves probability density function (PDF), spectral, and increment statistics on HW without additional stationary training samples or numerical-solver calls during sampler training.