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Statistical Error Bounds for Generative Solvers of Chaotic PDEs: Wasserstein Stability, Generalization, and Turbulence

Victor Armegioiu

TL;DR

This work develops a law-level analysis compatible with the correlation-measure framework of Lanthaler--Mishra--Par\'es-Pulido (LM), and shows how common finite-grid diagnostics--proper distributional scores and likelihood-style certificates--admit principled interpretations as resolved observables within the same statistical-solution framework.

Abstract

Statistical solutions of incompressible Euler describe turbulent dynamics as time-parameterized laws on $L^2$ whose multi-point correlations satisfy an infinite hierarchy of weak identities. Modern generative samplers for PDE forecasting (flow matching, rectified flows, diffusion via probability-flow ODEs) are measure-transport mechanisms and therefore induce Markov operators on laws. We develop a law-level analysis compatible with the correlation-measure framework of Lanthaler--Mishra--Parés-Pulido (LM): convergence in $d_T(μ,ν)=\int_{0}^{T} W_{1}\!\bigl(μ_t,ν_t\bigr)\,\mathrm{d}t$, compactness controlled by structure functions, and identification of limits through hierarchy identities. Quantitatively, we prove a $W_2$ stability estimate whose growth rate is a distance-weighted average strain under optimal couplings, and a one-step error decomposition into a resolved mismatch term and an unavoidable high-frequency coverage tail controlled by structure-function (spectral) bounds. These inputs propagate through multi-step rollouts via a discrete Grönwall recursion with amplification governed by the average-strain exponent rather than a worst-case Lipschitz constant. On the qualitative side, sampler-native path controls yield LM time regularity; together with uniform energy and structure-function bounds this gives precompactness in $d_T$ and strong convergence of LM-admissible observables. If hierarchy residuals vanish along a sequence, every limit is an LM statistical solution, with residuals bounded by training-native drift/score regression errors. Finally, we show how common finite-grid diagnostics--proper distributional scores and likelihood-style certificates--admit principled interpretations as resolved observables within the same statistical-solution framework.

Statistical Error Bounds for Generative Solvers of Chaotic PDEs: Wasserstein Stability, Generalization, and Turbulence

TL;DR

This work develops a law-level analysis compatible with the correlation-measure framework of Lanthaler--Mishra--Par\'es-Pulido (LM), and shows how common finite-grid diagnostics--proper distributional scores and likelihood-style certificates--admit principled interpretations as resolved observables within the same statistical-solution framework.

Abstract

Statistical solutions of incompressible Euler describe turbulent dynamics as time-parameterized laws on whose multi-point correlations satisfy an infinite hierarchy of weak identities. Modern generative samplers for PDE forecasting (flow matching, rectified flows, diffusion via probability-flow ODEs) are measure-transport mechanisms and therefore induce Markov operators on laws. We develop a law-level analysis compatible with the correlation-measure framework of Lanthaler--Mishra--Parés-Pulido (LM): convergence in , compactness controlled by structure functions, and identification of limits through hierarchy identities. Quantitatively, we prove a stability estimate whose growth rate is a distance-weighted average strain under optimal couplings, and a one-step error decomposition into a resolved mismatch term and an unavoidable high-frequency coverage tail controlled by structure-function (spectral) bounds. These inputs propagate through multi-step rollouts via a discrete Grönwall recursion with amplification governed by the average-strain exponent rather than a worst-case Lipschitz constant. On the qualitative side, sampler-native path controls yield LM time regularity; together with uniform energy and structure-function bounds this gives precompactness in and strong convergence of LM-admissible observables. If hierarchy residuals vanish along a sequence, every limit is an LM statistical solution, with residuals bounded by training-native drift/score regression errors. Finally, we show how common finite-grid diagnostics--proper distributional scores and likelihood-style certificates--admit principled interpretations as resolved observables within the same statistical-solution framework.
Paper Structure (95 sections, 46 theorems, 335 equations)

This paper contains 95 sections, 46 theorems, 335 equations.

Table of Contents

  1. Introduction
  2. What is missing in current ML practice.
  3. Why the Lanthaler--Mishra--Parés-Pulido framework.
  4. Samplers are law operators.
  5. Two threads, one interface: quantitative bounds and statistical-solution identification.
  6. Scope and reference dynamics.
  7. Contributions.
  8. Motivation and related work
  9. The LM framework: laws, observables, compactness, statistical solutions
  10. Basic notation and phase space
  11. Wasserstein distances
  12. $L^1_t(\mathcal{P})$ and the LM metric $d_T$
  13. Time-regularity
  14. Structure functions and a pointwise-to-time-averaged link
  15. Correlation measures and incompressibility (LM Definition 2.3 and Lemma 3.1)
  16. ...and 80 more sections

Key Result

Lemma 2.5

Let $\mu^m_\cdot\in L^1([0,T);\mathcal{P}(L^2_x))$ be time-regular with the same constants $(C,L)$ in the sense of Definition def:time-regular. Assume Assume $D=\mathbb{T}^d$ so that the embedding $L^2_x\hookrightarrow H^{-L}(D)$ is compact. Then $\mu_\cdot$ is time-regular with the same $(C,L)$.

Theorems & Definitions (115)

  • Definition 2.1: $L^1_t(\mathcal{P})$ and $d_T$
  • Remark 2.2: Why $L^1_t(\mathcal{P})$ is the right ambient space
  • Definition 2.3: Time-regularity
  • Remark 2.4: How time-regularity will be verified later
  • Lemma 2.5: LM time-regularity is closed under $d_T$ limits
  • proof
  • Lemma 2.6: Pointwise structure modulus implies the LM time-averaged bound
  • proof
  • Definition 2.7: LM-admissible observables
  • Lemma 2.8: Marginalization of correlation measures
  • ...and 105 more