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

arXiv 2605.21388 ↗First posted 20 May 2026cs.LG cs.AI cs.NAcited by 0

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Abstract (arXiv)

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.

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References · 20

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,636
  5. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  8. 2022Convergence of denoising diffusion models under the manifold hypothesiscited by 242
  9. 2019Optimal transport mapping via input convex neural networkscited by 206
  10. 2019Deep Network Approximation Characterized by Number of Neuronscited by 122
  11. 2018How Well Generative Adversarial Networks Learn Distributionscited by 78
  12. 2020Some Theoretical Insights into Wasserstein GANscited by 28
8 more
  1. 2020Approximating smooth functions by deep neural networks with sigmoid activation functioncited by 18
  2. 2024On the Statistical Properties of Generative Adversarial Models for Low Intrinsic Data Dimensioncited by 11
  3. 2024On the rates of convergence for learning with convolutional neural networkscited by 10
  4. 2024Wasserstein Bounds for generative diffusion models with Gaussian tail targetscited by 10
  5. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  6. 2025Pathway to O(√d) Complexity bound under Wasserstein metric of flow-based modelscited by 4
  7. 2022A DeepParticle method for learning and generating aggregation patterns in multi-dimensional Keller-Segel chemotaxis systemscited by 2
  8. 2022A variational neural network approach for glacier modelling with nonlinear rheologycited by 2

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