On the Regularity and Generalization of One-Step Wasserstein-guided Generative Models for PDE-Induced Measures

TL;DR

This paper provides a theoretical framework for understanding the regularity and generalization of one-step Wasserstein-guided generative models for PDE-induced measures, with proofs of regularity and bounds on approximation error.

Contribution

It establishes regularity results for transport maps in PDE-induced measures and offers a theoretical justification for single-step generative models like DeepParticle.

Findings

Target measures satisfy doubling conditions.

Optimal transport maps are Hölder continuous.

Derived excess-risk bounds for DeepParticle.

Abstract

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 H\"older 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.