Macrocanonical Generator Networks: data-efficient neural surrogates for amortized physics simulation

Many scientific simulations are computationally expensive, limiting their use in simulation-based inference, uncertainty quantification, and decision making. Resorting to imperfect simulators leads to bias, which may be avoidable. We introduce *macrocanonical generator networks*, a framework for learning fast, data-efficient neural surrogates for amortized simulation of stationary physical processes by matching multiscale statistics of a target process. Inspired by microcanonical maximum-entropy synthesis, which matches feature statistics per sample by inner-loop optimization, our approach instead trains a neural generator to satisfy the same constraints in expectation, preserving physically realistic sample-to-sample fluctuations. We prove that gradient-descent training induces preconditioned gradient-descent dynamics in sample space that inherit the symmetry-preservation properties of the microcanonical gradient descent of Bruna and Mallat, amortizing its per-sample inner-loop optimization into a single training run. We further show that residual generator parameterizations initialized near the identity, trained with a small learning rate, preserve entropy locally during early optimization and mitigate mode collapse. Experiments on cosmology and fluid simulations show that macrocanonical generators recover realistic samples from extremely small training sets, in some cases a single training example, while generating new realizations more than $10^6$ times faster per sample than the reference simulator and more than $10^3$ times faster than microcanonical gradient descent. The trained generators can be reused for downstream pretraining, including simulation-based inference, providing a data-efficient route to multi-fidelity scientific machine learning workflows.