arxiv: 2605.13790 · v1 · submitted 2026-05-13 · 💻 cs.LG · cs.AI

Recognition: no theorem link

Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations

Zhonghao Li , Chaoyu Liu , Qian Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:03 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords PDE solvingsparse observationslatent diffusionneural solverdenoisingvariational autoencoderbidirectional learningsuper-resolution

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The pith

Di-BiLPS solves both forward and inverse PDE problems from as little as 3 percent sparse observations by operating entirely in a compressed latent space.

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

The paper presents Di-BiLPS as a single neural architecture that addresses the challenge of solving partial differential equations when only a tiny fraction of the data is available. It first maps high-dimensional inputs to a compact latent representation with a variational autoencoder, then applies contrastive learning and a latent diffusion process to capture uncertainty and align features. A specialized denoising step guided by the PDE itself then refines the latent solution. This design keeps inference fast at high resolutions and supports direct prediction over continuous domains, delivering state-of-the-art accuracy on standard PDE benchmarks even when observations fall to 3 percent density.

Core claim

Di-BiLPS is a bidirectional latent-PDE-solver that integrates a variational autoencoder for dimensionality reduction, a latent diffusion module for uncertainty modeling, contrastive representation alignment, and a PDE-informed denoising algorithm based on a variance-preserving diffusion process, allowing accurate forward and inverse solutions under extremely sparse inputs while enabling zero-shot super-resolution over continuous spatial-temporal domains.

What carries the argument

The PDE-informed denoising algorithm that runs inside the learned latent space of the bidirectional solver, recovering physics-consistent fields directly from sparse observations without repeated high-dimensional operations.

If this is right

The same trained model handles both forward simulation and inverse parameter recovery for the tested PDE families.
Inference cost drops substantially compared with existing neural PDE methods at equivalent resolution.
Predictions can be made at any continuous spatial or temporal location without retraining or interpolation steps.
Performance remains state-of-the-art across multiple benchmarks down to 3 percent observation density.

Where Pith is reading between the lines

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

The latent-space approach could be tested on PDEs with stochastic terms or discontinuous solutions to check whether the denoising step still preserves physical invariants.
Pairing the framework with adaptive sampling of observation locations might further lower the required density below 3 percent in practice.
The continuous-domain output suggests direct use in real-time control loops where sensor data arrives irregularly.

Load-bearing premise

The denoising process inside the latent space recovers the true physics of the original PDE without introducing systematic bias or artifacts when input observations drop to 3 percent or fewer.

What would settle it

Apply Di-BiLPS to a standard PDE benchmark at precisely 3 percent random sparse observations and measure whether the recovered solution deviates from the known ground truth by more than classical numerical solvers or exhibits non-physical features such as artificial oscillations.

Figures

Figures reproduced from arXiv: 2605.13790 by Chaoyu Liu, Qian Zhang, Zhonghao Li.

**Figure 1.** Figure 1: Overview of the architecture of Di-BiLPS : (a) Compression module via a pre-trained VAE (left), (b) Diffusion module with proposed PDE-Guided denoising algorithm(middle), and (c) Contrastive learning module (right). Proposed design leverages module (a) to extract informative and robust representations from extremely sparse inputs and performs high-speed denoising algorithm within the compressed latent spac… view at source ↗

**Figure 2.** Figure 2: Illustration of contrastive learning module. Proposed GINO-ViT framework encodes both sparse and full observations into a unified latent space. By optimizing the match score within this space, framework is designed to capture informative and robust representations between sparse and full observations. Here, we emphasize that ViT modules with the same color share weights in this figure. where modules GINOE… view at source ↗

**Figure 3.** Figure 3: Comparison between our model and DiffusionPDE on the Darcy Flow (top) and inhomogeneous Helmholtz equation (bottom) tasks. 5.3. Main Evaluation Results [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: Comparison among our model, GraphPDE and DiffusionPDE(Huang et al., 2024) on the inverse problem of Bounded Navier-Stokes equations [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 4.** Figure 4: Comparison of all baseline models(Huang et al., 2024) listed in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 6.** Figure 6: Zero-shot super-resolution results on the forward problem of the inhomogeneous Helmholtz equations. and inverse settings. In this table, PDE Guidance corresponds to the term Npde as defined in Eq. (15), Observation Guidance refers to Nobs defined in Eq. (17), and Condition Guidance denotes the conditional input C to the noise estimation network ϵθ. Removing the PDE guidance results in negligible performa… view at source ↗

read the original abstract

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

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

Di-BiLPS packages VAE compression, latent diffusion denoising, and contrastive alignment into a bidirectional solver that targets extreme sparsity in PDEs, with plausible efficiency gains if the latent physics enforcement holds after decoding.

read the letter

Di-BiLPS compresses high-dimensional PDE fields with a VAE, runs a variance-preserving diffusion process in that latent space with an added PDE-informed denoising step, and uses contrastive alignment to support both forward and inverse tasks under very low observation density. The main practical claim is that this stays accurate at 3% sparsity on standard benchmarks, cuts compute relative to grid-based alternatives, and supports zero-shot super-resolution over continuous domains. Those are the pieces worth noting first. The bidirectional framing and the decision to keep everything in latent space are reasonable engineering choices that address real bottlenecks in sensor-limited settings. The experiments appear to show consistent outperformance on the usual test problems, which is the concrete evidence the paper supplies. The soft spot is the physics fidelity after decoding. The VAE is trained without an explicit residual term, and the denoising operates on compressed codes, so nothing in the construction automatically guarantees that the decoded output satisfies the original PDE to high accuracy once observations fall to 3%. If the full paper only reports benchmark error without residual checks or ablations that isolate the PDE-informed term, that gap remains. This is aimed at groups already working on neural PDE solvers who need to handle patchy data. Readers who care about implementation details for latent diffusion on scientific problems will get value from the method description and numbers. It deserves a serious referee because the empirical results are presented across multiple benchmarks and the architecture is spelled out clearly, even though the central claim on latent-space physics preservation is still empirical rather than guaranteed.

Referee Report

3 major / 2 minor

Summary. The paper proposes Di-BiLPS, a unified neural framework for forward and inverse PDE problems under extremely sparse observations (down to 3%). It combines a VAE to compress inputs into latent space, a latent diffusion module with variance-preserving denoising, contrastive learning for alignment, and a PDE-informed denoising step. The central claims are consistent SOTA benchmark performance, substantially reduced computational cost, and zero-shot super-resolution over continuous spatio-temporal domains.

Significance. If the claims hold, the work would be significant for physics-informed machine learning by demonstrating that latent-space diffusion can handle extreme sparsity while enabling efficient inference and continuous-domain prediction, addressing a key limitation of existing neural PDE solvers.

major comments (3)

[§3.3] §3.3 (PDE-informed denoising): The variance-preserving diffusion process is defined entirely in latent space with no explicit residual term or constraint linking back to the original PDE; the manuscript provides no derivation or bound showing that decoded outputs satisfy the PDE residual to high accuracy at 3% sparsity, which is load-bearing for the central claim.
[§5] §5 (Experiments): Reported SOTA results lack error bars, standard deviations across runs, or statistical significance tests; without these, the magnitude of improvement over baselines cannot be assessed reliably, especially given the skeptic's concern about artifacts at low sparsity.
[Table 1] Table 1 / §5.2: Ablation studies on component contributions (VAE compression, diffusion schedule, contrastive loss) are not reported at the 3% sparsity level; this leaves open whether the PDE-informed step is actually responsible for the claimed recovery or whether performance relies on the VAE alone.

minor comments (2)

[§3.2] Notation for the latent diffusion variance schedule is introduced in §3.2 but not consistently used in the algorithm pseudocode, making the implementation details harder to follow.
[Figure 3] Figure 3 caption does not specify the exact sparsity percentages shown in the qualitative results, reducing reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and will revise the manuscript to strengthen the theoretical grounding, experimental reporting, and ablation analysis.

read point-by-point responses

Referee: [§3.3] §3.3 (PDE-informed denoising): The variance-preserving diffusion process is defined entirely in latent space with no explicit residual term or constraint linking back to the original PDE; the manuscript provides no derivation or bound showing that decoded outputs satisfy the PDE residual to high accuracy at 3% sparsity, which is load-bearing for the central claim.

Authors: We agree that an explicit derivation linking the latent-space variance-preserving process to the decoded PDE residual is currently missing and is important for the central claim. In the revision we will add a new subsection deriving the consistency of the latent diffusion step with the original PDE residual (via the VAE decoder and the contrastive alignment), together with empirical residual bounds computed on the benchmark problems at 3% sparsity. This will clarify how the PDE-informed denoising enforces the governing equations without an explicit residual term in latent space. revision: yes
Referee: [§5] §5 (Experiments): Reported SOTA results lack error bars, standard deviations across runs, or statistical significance tests; without these, the magnitude of improvement over baselines cannot be assessed reliably, especially given the skeptic's concern about artifacts at low sparsity.

Authors: We acknowledge the absence of error bars and statistical tests. In the revised manuscript we will report mean performance and standard deviation over five independent runs for all main results, and we will add paired t-tests (or Wilcoxon tests where appropriate) to establish statistical significance of the improvements over baselines at the 3% sparsity level. revision: yes
Referee: [Table 1] Table 1 / §5.2: Ablation studies on component contributions (VAE compression, diffusion schedule, contrastive loss) are not reported at the 3% sparsity level; this leaves open whether the PDE-informed step is actually responsible for the claimed recovery or whether performance relies on the VAE alone.

Authors: The existing ablations were performed at moderate sparsity to isolate component effects cleanly. We will extend Table 1 (and the corresponding discussion in §5.2) with a new set of ablations conducted specifically at the 3% sparsity level, showing the incremental contribution of the PDE-informed denoising step, the diffusion schedule, and the contrastive loss when observations are extremely sparse. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on empirical benchmarks

full rationale

The paper introduces Di-BiLPS as a composite framework (VAE compression + latent diffusion + PDE-informed denoising) and supports its SOTA claims exclusively through reported experimental results on multiple PDE benchmarks at low sparsity levels. No equations are presented that define a target quantity in terms of itself, no fitted parameters are relabeled as predictions, and no load-bearing self-citations or uniqueness theorems are invoked to close the derivation. The performance assertions remain externally falsifiable via the stated benchmarks rather than reducing to internal construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim depends on the latent space preserving PDE dynamics sufficiently for the denoising process to succeed; this is an unproven domain assumption rather than a derived result.

free parameters (2)

latent dimension
Size of the compressed representation chosen to balance fidelity and efficiency; value not specified in abstract.
diffusion variance schedule
Parameters controlling the variance-preserving diffusion process, tuned for the PDE denoising task.

axioms (1)

domain assumption The learned latent space preserves the essential dynamics of the original PDE
Invoked to justify that denoising in latent space yields accurate physical predictions.

pith-pipeline@v0.9.0 · 5520 in / 1279 out tokens · 43685 ms · 2026-05-14T19:03:05.072216+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 16 canonical work pages · 6 internal anchors

[1]

World Simulation with Video Foundation Models for Physical AI

Ali, A., Bai, J., Bala, M., Balaji, Y., Blakeman, A., Cai, T., Cao, J., Cao, T., Cha, E., Chao, Y.-W., et al. World simulation with video foundation mod- els for physical ai. arXiv preprint arXiv:2511.00062 ,

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Universal physics Transformers

Alkin, B., Fürst, A., Schmid, S., Gruber, L., Holzleit- ner, M., and Brandstetter, J. Universal physics Transformers. arXiv preprint arXiv:2402.12365 ,

work page arXiv
[3]

F., Stuart, A., Mahoney, M

Cheng, C., Han, B., Maddix, D., Ansari, A. F., Stuart, A., Mahoney, M. W., and Wang, B. Gradient-free generation for hard-constrained systems. In Interna- tional Conference on Learning Representations , vol- ume 2025, pp. 100510–100539,

2025
[4]

Diffusion Posterior Sampling for General Noisy Inverse Problems

Chung, H., Kim, J., Mccann, M. T., Klasky, M. L., and Ye, J. C. Diffusion posterior sampling for general noisy inverse problems. arXiv preprint arXiv:2209.14687,

work page internal anchor Pith review Pith/arXiv arXiv
[5]

An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale

Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weis- senborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929,

work page internal anchor Pith review Pith/arXiv arXiv 2010
[6]

Huang, J., Yang, G., Wang, Z., and Park, J

URL https://arxiv.org/abs/2302.14376. Huang, J., Yang, G., Wang, Z., and Park, J. J. Dif- fusionpde: Generative pde-solving under partial ob- servation. arXiv preprint arXiv:2406.17763 ,

work page arXiv
[7]

Perceptual Losses for Real-Time Style Transfer and Super-Resolution

URL https://arxiv.org/abs/ 1603.08155. Karumuri, S., Graham-Brady, L., and Goswami, S. Physics-informed latent neural operator for real- time predictions of time-dependent parametric pdes. Computer Methods in Applied Mechanics and Engi- neering, 450:118599,

work page internal anchor Pith review Pith/arXiv arXiv
[8]

Neural Operator: Graph Kernel Network for Partial Differential Equations

Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandku- mar, A. Neural operator: graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485, 2020a. Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier neural operator for parametric ...

work page internal anchor Pith review arXiv 2003
[9]

Scal- able Transformer for PDE surrogate modeling

10 Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations Li, Z., Shu, D., and Barati Farimani, A. Scal- able Transformer for PDE surrogate modeling. Ad- vances in Neural Information Processing Systems (NeurIPS), 2024a. Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., and Anandkumar, A. Physi...

work page arXiv 2025
[10]

doi: 10.1063/5.0188170

ISSN 1070-6631. doi: 10.1063/5.0188170. URL https://doi.org/ 10.1063/5.0188170. Molinaro, R., Lanthaler, S., Raonić, B., Rohner, T., Armegioiu, V., Simonis, S., Grund, D., Ramic, Y., Wan, Z. Y., Sha, F., et al. Generative ai for fast and accurate statistical computation of fluids. arXiv preprint arXiv:2409.18359 ,

work page doi:10.1063/5.0188170
[11]

U-net: Convolutional networks for biomedical image seg- mentation

Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image seg- mentation. Medical image computing and computer- assisted intervention–MICCAI 2015: 18th interna- tional conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18 , pp. 234–241,

2015
[12]

Denoising Diffusion Implicit Models

Song, J., Meng, C., and Ermon, S. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502 , 2020a. Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456 , 2020b. Team, M. L., Cai, X., Huang, Q., Kang, Z., Li, H...

work page internal anchor Pith review Pith/arXiv arXiv 2010
[13]

DexVLA: Vision-Language Model with Plug-In Diffusion Expert for General Robot Control

Wen, J., Zhu, Y., Li, J., Tang, Z., Shen, C., and Feng, F. Dexvla: Vision-language model with plug- in diffusion expert for general robot control. arXiv preprint arXiv:2502.05855 ,

work page Pith review arXiv
[14]

Wight, C. L. and Zhao, J. Solving allen-cahn and cahn-hilliard equations using the adaptive physics informed neural networks. arXiv preprint arXiv:2007.04542,

work page arXiv 2007
[15]

Solving high-dimensional PDEs with latent spectral models

Wu, H., Hu, T., Luo, H., Wang, J., and Long, M. Solving high-dimensional PDEs with latent spectral models. arXiv preprint arXiv:2301.12664 ,

work page arXiv
[16]

Transolver: A fast transformer solver for pdes on general geometries.arXiv preprint arXiv:2402.02366, 2024

URL https://arxiv. org/abs/2402.02366. Xiao, Z., Hao, Z., Lin, B., Deng, Z., and Su, H. Im- proved operator learning by orthogonal attention. arXiv preprint arXiv:2310.12487 ,

work page arXiv
[17]

B., and Wetzstein, G

Zhao, Q., Lindell, D. B., and Wetzstein, G. Learn- ing to solve pde-constrained inverse problems with graph networks. arXiv preprint arXiv:2206.00711 ,

work page arXiv
[18]

Diffusion-based planning for autonomous driving with flexible guidance.arXiv preprint arXiv:2501.15564,

Zheng, Y., Liang, R., Zheng, K., Zheng, J., Mao, L., Li, J., Gu, W., Ai, R., Li, S. E., Zhan, X., et al. Diffusion-based planning for autonomous driving with flexible guidance. arXiv preprint arXiv:2501.15564,

work page arXiv
[19]

Related Work A.1

12 Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations A. Related Work A.1. Operator Learning Operator learning aims to approximate mappings between infinite dimensional function spaces, typically from coeﬀicients, source terms, or initial conditions to the corresponding solution functions. Instead of explicitly solv- ing...

2021
[20]

, 2008), enabling more compact latent representations

introduces additional encoding and decoding objectives and compresses geometric information into super-nodes with graph neural networks ( Scarselli et al. , 2008), enabling more compact latent representations. In contrast to these approaches, our method learns the bidirectional mapping through latent representations in a fully end-to-end manner. Leveragin...

2008
[21]

The geometry includes randomly sized cylindrical obstacles, with a turbulent inflow driven from the upper boundary. No-slip conditions 14 Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations (v = 0 ) are strictly enforced on the lateral walls ( ∂Ω left, ∂ Ω right) and the boundaries of the internal cylinders (∂Ω cylinder)....

2023