Why one-step generative models work for physics-based probability distributions

Generative models excel empirically but lack theory for scientific computing tasks like sampling from PDE solutions. This work proves that probability measures arising from PDEs have special structure—they satisfy doubling conditions—which guarantees smooth transport maps between simple distributions and complex PDE-induced ones. This regularity justifies training a single neural network to learn the transport map, and the authors derive concrete error bounds for the DeepParticle model showing convergence rates.