arxiv: 2509.20098 · v2 · submitted 2025-09-24 · 💻 cs.LG

Incomplete Data, Complete Dynamics: A Diffusion Approach

Zihan Zhou , Chenguang Wang , Hongyi Ye , Yongtao Guan , Tianshu Yu This is my paper

Pith reviewed 2026-05-18 14:14 UTC · model grok-4.3

classification 💻 cs.LG

keywords diffusion modelsincomplete dataphysical dynamicsdata imputationconditional generationasymptotic convergenceirregular samplingmachine learning for science

0 comments

The pith

Diffusion training on incomplete data converges asymptotically to the true complete generative process for physical dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a diffusion-based framework that learns physical dynamics directly from incomplete and irregularly sampled observations. Each incomplete sample is partitioned into an observed context and an unobserved query via a designed splitting strategy, after which a conditional diffusion model is trained to reconstruct the missing query. Theoretical analysis shows that this training paradigm converges to the true underlying complete generative process under mild regularity conditions. This matters for real-world settings such as fluid flows or weather data, where full observations are rarely available, because the method enables accurate imputation and dynamics modeling without requiring complete-data supervision.

Core claim

The central claim is that the diffusion training paradigm on incomplete data achieves asymptotic convergence to the true complete generative process under mild regularity conditions. This is realized by strategically partitioning each incomplete sample into observed context and unobserved query components and training a conditional diffusion model to reconstruct the missing query portions given the available contexts, thereby enabling accurate imputation across arbitrary observation patterns without complete data supervision.

What carries the argument

The splitting strategy that partitions each incomplete sample into observed context and unobserved query components, allowing the conditional diffusion model to reconstruct missing portions and recover the full dynamics.

If this is right

Accurate imputation is possible across arbitrary observation patterns without requiring complete data supervision.
The method significantly outperforms existing baselines on synthetic and real-world physical dynamics benchmarks including fluid flows and weather systems.
Particularly strong performance holds in limited and irregular observation regimes.
The approach supplies a theoretically principled route to learning and imputing partially observed dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The convergence guarantee may extend to other generative modeling families if an analogous conditional splitting can be defined.
This suggests incomplete observational streams could suffice for online or continual learning of dynamics in sensor networks.
Integration with physics-informed regularizers might further stabilize the recovered dynamics on very sparse real-world data.
The framework could be tested on non-physical systems such as epidemiological or financial time series that share irregular sampling traits.

Load-bearing premise

The carefully designed splitting strategy that partitions each incomplete sample into observed context and unobserved query components is sufficient to enable the conditional diffusion model to recover the full underlying dynamics.

What would settle it

Training the model on a synthetic complete dataset with known dynamics and then checking whether the learned conditional distribution matches the true generative process when queries are masked according to the splitting rule; systematic mismatch on held-out complete trajectories would falsify the asymptotic convergence result.

Figures

Figures reproduced from arXiv: 2509.20098 by Chenguang Wang, Hongyi Ye, Tianshu Yu, Yongtao Guan, Zihan Zhou.

**Figure 3.** Figure 3: Comparison of original and imputed data from the Navier-Stokes dataset (60% observed [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 2.** Figure 2: Data imputation visualization Shallow Water and Advection Equations. We consider two fundamental geophysical PDE systems: the shallow water equations governing fluid dynamics with rotation, and the linear advection equation describing scalar transport. Each dataset contains 5k training, 1k validation, and 1k test samples with 32 × 32 spatial resolution and 50 temporal frames, generated with randomized ph… view at source ↗

**Figure 4.** Figure 4: Imputed results on 2D Shallow Water dataset where 30% of the original data points are [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗

**Figure 5.** Figure 5: Imputed results on 2D Advection dataset where 30% of the original data points are avail [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Sample imputation results on 2D Navier-Stokes dataset where 80% of the original data [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗

**Figure 7.** Figure 7: Sample imputation results on 2D Navier-Stokes dataset where 60% of the original data [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: Sample imputation results on 2D Navier-Stokes dataset where 20% of the original data [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗

**Figure 9.** Figure 9: Comparison of original and imputed data from the Navier-Stokes dataset (one missing [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗

**Figure 10.** Figure 10: Imputation results for our method vs. MissDiff. The imputed block is the center one. [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗

**Figure 11.** Figure 11: Imputation results on the ERA5 dataset with 20% observed points. Each subfig [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗

read the original abstract

Learning physical dynamics from data is a fundamental challenge in machine learning and scientific modeling. Real-world observational data are inherently incomplete and irregularly sampled, posing significant challenges for existing data-driven approaches. In this work, we propose a principled diffusion-based framework for learning physical systems from incomplete training samples. To this end, our method strategically partitions each such sample into observed context and unobserved query components through a carefully designed splitting strategy, then trains a conditional diffusion model to reconstruct the missing query portions given available contexts. This formulation enables accurate imputation across arbitrary observation patterns without requiring complete data supervision. Specifically, we provide theoretical analysis demonstrating that our diffusion training paradigm on incomplete data achieves asymptotic convergence to the true complete generative process under mild regularity conditions. Empirically, we show that our method significantly outperforms existing baselines on synthetic and real-world physical dynamics benchmarks, including fluid flows and weather systems, with particularly strong performance in limited and irregular observation regimes. These results demonstrate the effectiveness of our theoretically principled approach for learning and imputing partially observed dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper's main contribution is a partitioning strategy for training conditional diffusion models on incomplete physical trajectories, with a claimed asymptotic convergence to the full generative process, though the theory needs close checking.

read the letter

The central idea here is splitting each incomplete sample into an observed context and an unobserved query, then training a conditional diffusion model to fill in the missing parts. This is meant to let the model learn full dynamics without ever seeing complete trajectories during training, and the authors claim it converges asymptotically to the true complete process under mild regularity conditions. That framing is the main thing to know up front. Empirically the method is tested on synthetic data plus real fluid flows and weather systems, and it beats baselines especially when observations are sparse or irregular. That practical focus on scientific data with missing entries is where the work lands most cleanly. The partitioning-plus-conditional-diffusion combination for dynamics imputation looks like the distinct piece relative to standard diffusion setups. The experiments give some evidence that the approach works in limited-data regimes, which is useful for the target applications. The soft spot is the theoretical claim. The convergence argument rests on the splitting operator producing a training distribution whose conditional score-matching loss recovers the full joint, yet the abstract leaves the precise regularity conditions and measure-theoretic requirements unspecified. The stress-test note correctly flags that if the split does not induce the right equivalence in the limit, the model could converge only to marginals or conditionals that do not determine the underlying dynamics. Without the full derivations it is difficult to judge how tight that step is. This is aimed at people working on scientific machine learning and data-driven modeling of physical systems who routinely deal with incomplete observations. A reader looking for imputation methods in dynamical systems or diffusion applications to science would find the experiments and framework worth examining. It deserves a serious referee because the problem is real, the empirical results are presented on relevant benchmarks, and there is an attempt at theory, even if that theory will likely require clarification and possibly strengthening during review. I would send it out for peer review with instructions to focus on the convergence argument and the exact properties required of the splitting strategy.

Referee Report

2 major / 2 minor

Summary. The paper introduces a diffusion-based framework for learning physical dynamics from incomplete and irregularly sampled data. Each incomplete sample is partitioned into observed context and unobserved query components using a designed splitting strategy. A conditional diffusion model is trained to reconstruct the query from the context, enabling imputation for arbitrary observation patterns. The authors provide theoretical analysis showing asymptotic convergence to the true complete generative process under mild regularity conditions and demonstrate empirical outperformance over baselines on synthetic and real-world benchmarks such as fluid flows and weather systems.

Significance. If the theoretical result holds, this work would offer a significant advance in data-driven modeling of physical systems by handling incomplete observations without requiring complete data. The approach could improve accuracy in scientific applications like fluid dynamics and meteorology where data is often partial. The empirical results indicate particular strength in limited and irregular observation settings, which are common in practice.

major comments (2)

[Theoretical Analysis] The central convergence claim relies on the splitting strategy inducing a training distribution where the conditional diffusion loss recovers the full joint score function. However, the manuscript treats the splitting operator as given without deriving the required measure-theoretic or measure-preserving properties for equivalence to score matching on full trajectories. This is load-bearing for the asymptotic convergence result to the true complete generative process.
[§4.2] The mild regularity conditions under which convergence is claimed are not specified in detail; without explicit statement of these conditions and how they ensure the conditional model recovers the underlying dynamics, the support for the claim remains incomplete.

minor comments (2)

[Experiments] The description of the experimental protocols could be expanded to include more details on how the incomplete data is generated for the benchmarks.
[Notation] Some notation in the method section could be clarified for readers unfamiliar with conditional diffusion models.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the two major comments below and will revise the manuscript to provide the requested clarifications and derivations in the theoretical analysis.

read point-by-point responses

Referee: [Theoretical Analysis] The central convergence claim relies on the splitting strategy inducing a training distribution where the conditional diffusion loss recovers the full joint score function. However, the manuscript treats the splitting operator as given without deriving the required measure-theoretic or measure-preserving properties for equivalence to score matching on full trajectories. This is load-bearing for the asymptotic convergence result to the true complete generative process.

Authors: We appreciate the referee pointing out this key technical step. The current manuscript presents the high-level argument that the splitting induces the desired conditional training distribution but does not contain an explicit measure-theoretic derivation of the required properties. In the revision we will add a dedicated appendix subsection that formally shows the splitting operator is measure-preserving with respect to the data measure and that the conditional score-matching objective is equivalent to score matching on the full trajectories, thereby supporting the asymptotic convergence claim. revision: yes
Referee: [§4.2] The mild regularity conditions under which convergence is claimed are not specified in detail; without explicit statement of these conditions and how they ensure the conditional model recovers the underlying dynamics, the support for the claim remains incomplete.

Authors: We agree that the regularity conditions should be stated more explicitly. In the revised §4.2 we will enumerate the precise assumptions (Lipschitz continuity of the vector field, standard Gaussian noise schedule, finite second moments of the data distribution, and compactness of the observation mask set) and insert a short proof sketch showing how each condition guarantees that the conditional diffusion process converges in distribution to the true complete generative process. revision: yes

Circularity Check

0 steps flagged

No circularity: theoretical convergence claim rests on external regularity conditions and splitting strategy without reduction to inputs by construction.

full rationale

The paper's derivation presents a conditional diffusion model trained on context-query splits from incomplete samples, with a theoretical result claiming asymptotic convergence to the true generative process under mild regularity conditions. No equations or steps in the abstract or description reduce the claimed convergence to a fitted parameter, self-definition, or self-citation chain; the splitting strategy is introduced as a design choice enabling the conditional objective rather than being derived from the target result itself. The analysis is framed as independent of the specific fitted values and relies on stated external assumptions, making the chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or invented entities; the primary unstated element is the set of mild regularity conditions required for the convergence result.

axioms (1)

domain assumption mild regularity conditions
Invoked to guarantee asymptotic convergence of the diffusion training on incomplete data to the true complete generative process.

pith-pipeline@v0.9.0 · 5709 in / 1183 out tokens · 73455 ms · 2026-05-18T14:14:57.934566+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 (Optimal solution under context masking...): xθ(t,Mctx ⊙ xobs,t,Mctx) = E[x0 | Mctx ⊙ xobs,t,Mctx] when union of query supports covers all dimensions.
Foundation.BranchSelection branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Principle 1 (Principle of uniform query exposure): P((Mqry)i=1 | Mctx) > 0 and approximately uniform for all unobserved dimensions.
Foundation.RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2 (Ensemble approximation convergence): lim K→∞ E[||μ̂K − E[x0|obs]||²] = ||E[E[x0|ctx]] − E[x0|obs] + E[b(ctx)]||²

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

work page
[54]

@esa (Ref

\@ifxundefined[1] #1\@undefined \@firstoftwo \@secondoftwo \@ifnum[1] #1 \@firstoftwo \@secondoftwo \@ifx[1] #1 \@firstoftwo \@secondoftwo [2] @ #1 \@temptokena #2 #1 @ \@temptokena \@ifclassloaded agu2001 natbib The agu2001 class already includes natbib coding, so you should not add it explicitly Type <Return> for now, but then later remove the command n...

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\@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@first@sw \@firstoftwo \@ifundefined NAT@b*@#2 \@firstoftwo @num @NAT@ctr \@secondoft...

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@open @close @open @close and [1] URL: #1 \@ifundefined chapter * \@mkboth \@ifxundefined @sectionbib * \@mkboth * \@mkboth\@gobbletwo \@ifclassloaded amsart * \@ifclassloaded amsbook * \@ifxundefined @heading @heading NAT@ctr thebibliography [1] @ \@biblabel @NAT@ctr \@bibsetup #1 @NAT@ctr @ @openbib .11em \@plus.33em \@minus.07em 4000 4000 `\.\@m @bibit...

work page doi:10.1016/j.ocemod.2018.09.006 2024