On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures
For researchers in scientific computing and generative modeling, this work offers rigorous theoretical guarantees for the regularity and generalization of one-step Wasserstein-guided generative models applied to PDE-induced measures, addressing a gap between empirical success and pessimistic theory.
This paper provides a theoretical framework proving that optimal transport maps from a uniform source to PDE-induced target measures are Hölder continuous, which justifies one-step generative models like DeepParticle. It derives excess-risk bounds and robustness estimates, with experiments supporting the theoretical rates.
Despite the remarkable empirical success of generative models, the available theory on their statistical accuracy in scientific computing remains largely pessimistic. This paper develops a theoretical framework for understanding the regularity of transport maps and the generalization properties of one-step Wasserstein-guided generative models for PDE-induced probability measures. We consider normalized target densities associated with linear elliptic and parabolic equations on bounded domains, as well as diffusion and Fokker--Planck equations on the torus. Under standard structural assumptions, we prove that these target measures satisfy doubling conditions. By combining this fact with regularity theory for optimal transport between doubling measures, we show that the optimal transport map from a uniform source measure to the target measure is Hölder continuous. This regularity yields an approximation-theoretic justification for one-step generative models that learn PDE-induced distributions via a single pushforward map. As a representative instance, we study DeepParticle and derive excess-risk bounds characterizing the discrepancy between the learned map and the population-optimal map. We also establish a robustness estimate under target shift and illustrate the theory with experiments which support the derived rates.