arxiv: 2604.27313 · v1 · submitted 2026-04-30 · 💻 cs.LG · cs.CV

Recognition: unknown

PINN-Cast: Exploring the Role of Continuous-Depth NODE in Transformers and Physics Informed Loss as Soft Physical Constraints in Short-term Weather Forecasting

Hira Saleem , Flora Salim , Cormac Purcell

Authors on Pith no claims yet

Pith reviewed 2026-05-07 09:59 UTC · model grok-4.3

classification 💻 cs.LG cs.CV

keywords transformerneural ordinary differential equationphysics informed neural networkweather forecastingcontinuous depthattention mechanismshort-term prediction

0 comments

The pith

A continuous-depth transformer encoder using Neural ODE dynamics and a physics-informed loss produces more accurate and physically consistent short-term weather forecasts than discrete transformer baselines.

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

The paper replaces discrete residual updates in transformer encoder blocks with Neural ODE dynamics solved by adaptive integration to model smooth latent processes in weather data. It adds an auxiliary derivative branch to the attention module for change-sensitive signals and trains with a customized loss that softly enforces physical consistency. If this holds, data-driven forecasters could close the gap with physics-based numerical weather prediction by respecting governing equations while retaining efficiency. A reader would care because operational forecasting demands both numerical skill and physical realism without prohibitive compute costs.

Core claim

The proposed PINN-Cast model integrates Neural ODE-based continuous updates inside each transformer encoder block in place of discrete residuals, pairs standard patch-wise self-attention with an auxiliary derivative operator on attention logits, and optimizes under a physics-informed objective; this combination yields forecasts that outperform both a standard discrete transformer and an earlier continuous-time Neural ODE forecaster on short-term weather tasks by capturing smoother dynamics and enforcing physical constraints as soft penalties.

What carries the argument

Neural ODE dynamics embedded as continuous-depth updates within transformer encoder blocks, solved via adaptive numerical integration to evolve representations, together with the two-branch attention module and the auxiliary physics-informed loss term.

If this is right

Forecasts respect governing physical principles more closely because the loss term acts as a soft constraint during training.
Latent representations evolve smoothly across depth rather than through abrupt discrete jumps, suiting continuous atmospheric processes.
The architecture remains compatible with standard transformer training pipelines while adding only adaptive ODE solvers and an extra attention branch.
Short-term prediction skill improves over both purely discrete and prior continuous-time baselines on the evaluated tasks.

Where Pith is reading between the lines

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

The same continuous-depth plus physics-loss pattern could transfer to other sequential physical domains such as fluid dynamics or climate modeling without redesigning the core blocks.
If the adaptive integration remains stable at longer horizons, the approach might reduce the need for separate physics post-processing steps in operational pipelines.
The derivative attention branch supplies an explicit sensitivity signal that might generalize to other tasks requiring detection of rapid changes in time series.

Load-bearing premise

That continuous ODE updates plus the derivative attention branch and physics loss will deliver higher forecast accuracy and physical consistency than discrete transformers without introducing instability or excessive computation.

What would settle it

A direct comparison on held-out weather datasets in which the PINN-Cast model shows no reduction in forecast error metrics or physical violation scores relative to the discrete transformer baseline.

Figures

Figures reproduced from arXiv: 2604.27313 by Cormac Purcell, Flora Salim, Hira Saleem.

**Figure 1.** Figure 1: Overall prediction pipeline. The model receives spatiotemporal weather view at source ↗

**Figure 2.** Figure 2: 12hr forecast results view at source ↗

**Figure 3.** Figure 3: 1 day forecast results 4.6 Ablation Studies We perform ablation studies to isolate the contribution of each proposed component to overall forecast accuracy. The full PINN-Cast model combines both architectural modifications with the physics-informed loss, allowing us to assess whether the improvements stem from richer attention interactions, smoother continuous-depth dynamics, or physics-motivated supervi… view at source ↗

**Figure 4.** Figure 4: Ablation studies highlight the results for different components of PINN view at source ↗

read the original abstract

Operational weather prediction has long relied on physics-based numerical weather prediction (NWP), whose accuracy comes at the cost of substantial compute and complex simulation workflows. Recent transformer-based forecasters offer efficient data-driven alternatives, however transformers are physics-agnostic models. Additionally, standard transformer encoders evolve representations through discrete layer updates that may be less suited to modeling smooth latent dynamics. In this work, we propose a continuous-depth transformer encoder for weather forecasting that integrates Neural Ordinary Differential Equation (Neural ODE) dynamics within each encoder block. Specifically, we replace discrete residual updates with ODE-based updates solved using adaptive numerical integration. We also introduce a two-branch attention module that combines conventional patch-wise self-attention with an auxiliary branch that applies a derivative operator to attention logits, providing an additional change-sensitive interaction signal. To further align forecasts with governing principles, we propose a customized physics-informed training objective that enforces physical consistency as a soft constraint. We evaluate the proposed method against a standard discrete transformer baseline and an existing continuous-time Neural ODE forecasting variant, demonstrating the importance of PINN-Cast in short term weather forecasting.

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

PINN-Cast puts Neural ODE dynamics inside transformer encoder blocks for weather forecasting plus a derivative attention branch and physics loss, but the abstract gives no numbers so the gains stay unproven.

read the letter

The paper's main move is to swap discrete residual updates in a transformer encoder for Neural ODE flows solved adaptively, add an auxiliary attention branch that differentiates the logits, and train with a custom physics-informed loss that acts as a soft constraint on weather variables. This targets the mismatch between standard transformers and the smooth evolution of atmospheric fields while trying to keep forecasts physically plausible without full NWP overhead. The design is a direct response to the discrete-layer limitation and the physics-agnostic nature of pure data-driven models. If the full experiments show stable integration and measurable lifts over both a plain transformer and an earlier Neural ODE forecaster, the combination could be worth trying in operational short-term setups. The motivation section lays out the rationale clearly and the two-branch attention is a concrete, if incremental, addition. The physics loss is customized rather than generic, which fits the weather domain. The abstract positions the work against relevant baselines, which is the right framing. The main soft spot is the complete absence of any quantitative results, error bars, dataset details, or ablation numbers in the provided text. Without those, it is impossible to tell whether the ODE replacement actually improves accuracy or physical consistency or whether it simply increases compute and risk of instability. The stress-test point about stiffness or excessive function evaluations in high-dimensional patch embeddings lands as a real concern; the abstract mentions adaptive integration but gives no solver type, tolerances, or regularization steps, so that needs checking in the methods. The approach builds on existing Neural ODE and PINN literature without obvious circularity. This is for groups already working on hybrid ML-physics forecasting who want to test continuous-depth encoders. A reader could pull the architectural sketch for their own experiments even if the results section turns out modest. Send it to peer review so the experiments, stability handling, and any ablations get proper scrutiny.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes PINN-Cast, a continuous-depth transformer encoder for short-term weather forecasting that integrates Neural ODE dynamics within each encoder block by replacing discrete residual updates with adaptive ODE integration. It further introduces a two-branch attention module combining standard self-attention with an auxiliary derivative operator applied to attention logits, and employs a customized physics-informed loss to enforce physical consistency as a soft constraint. The authors evaluate the approach against a standard discrete transformer baseline and an existing continuous-time Neural ODE variant, claiming it demonstrates the importance of these elements for improved forecasting accuracy and physical consistency.

Significance. If the empirical claims hold under rigorous testing, the work could meaningfully advance hybrid data-driven and physics-aware forecasting by embedding continuous latent dynamics and soft constraints into transformers, potentially yielding more stable and interpretable models than purely discrete architectures for evolving fields like weather. The combination of adaptive ODE flows with derivative-augmented attention represents a targeted architectural extension that addresses limitations of discrete layer stacking, though its practical gains depend on validation against established baselines with proper controls.

major comments (3)

[Abstract] Abstract: The central claim that the method 'demonstrates the importance of PINN-Cast in short term weather forecasting' is unsupported by any quantitative results, error metrics, baseline comparisons, error bars, or data-split descriptions. Without these, the superiority in accuracy and physical consistency cannot be assessed and the evaluation statement remains unverifiable.
[Abstract] Abstract (Proposed Method description): The integration of Neural ODE dynamics via adaptive numerical integration in each encoder block provides no details on the integrator type, tolerance settings, step-size adaptation, or mechanisms to detect or mitigate stiffness/divergence in high-dimensional weather patch embeddings. This is load-bearing for the continuous-depth claim, as instability or excessive NFE counts would invalidate the asserted advantages over discrete residuals.
[Abstract] Abstract (Attention module): The auxiliary derivative branch on attention logits is described without specifying the derivative computation method (e.g., automatic differentiation through the ODE solver or finite differences) or how its output is fused back into the dynamics and loss, leaving open questions about training stability and whether the 'change-sensitive interaction signal' actually contributes to the reported improvements.

minor comments (2)

[Abstract] Abstract: The phrase 'customized physics-informed training objective' is vague; specifying which physical principles (e.g., conservation laws or governing PDE residuals) are enforced would improve clarity even in the summary.
[Abstract] Abstract: Notation for the two-branch attention and ODE update could be introduced with a brief equation or reference to standard Neural ODE formulations to aid readers unfamiliar with the continuous-depth extension.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their insightful comments, which have helped us identify areas for improvement in the manuscript. We respond to each major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the method 'demonstrates the importance of PINN-Cast in short term weather forecasting' is unsupported by any quantitative results, error metrics, baseline comparisons, error bars, or data-split descriptions. Without these, the superiority in accuracy and physical consistency cannot be assessed and the evaluation statement remains unverifiable.

Authors: The abstract is intended as a concise overview. The full manuscript contains the requested quantitative results, including error metrics, baseline comparisons, error bars from repeated runs, and explicit data-split descriptions, all presented in the Experiments section with statistical significance testing. To address the concern directly, we will revise the abstract to incorporate key quantitative highlights and a brief reference to the evaluation protocol. revision: yes
Referee: [Abstract] Abstract (Proposed Method description): The integration of Neural ODE dynamics via adaptive numerical integration in each encoder block provides no details on the integrator type, tolerance settings, step-size adaptation, or mechanisms to detect or mitigate stiffness/divergence in high-dimensional weather patch embeddings. This is load-bearing for the continuous-depth claim, as instability or excessive NFE counts would invalidate the asserted advantages over discrete residuals.

Authors: We acknowledge the abstract's brevity on these implementation aspects. The manuscript specifies an adaptive ODE solver with defined tolerances, step-size adaptation, and stability monitoring via NFE counts to handle potential stiffness in the high-dimensional embeddings. We will revise the abstract to include a concise description of the integrator, tolerances, adaptation mechanism, and stability safeguards. revision: yes
Referee: [Abstract] Abstract (Attention module): The auxiliary derivative branch on attention logits is described without specifying the derivative computation method (e.g., automatic differentiation through the ODE solver or finite differences) or how its output is fused back into the dynamics and loss, leaving open questions about training stability and whether the 'change-sensitive interaction signal' actually contributes to the reported improvements.

Authors: The manuscript details that the derivative is obtained via automatic differentiation through the ODE solver and that the resulting features are fused by concatenation prior to the feed-forward network, with an auxiliary term in the loss to promote change sensitivity. Ablation experiments confirm the contribution of this branch to performance gains and training stability. We will update the abstract to specify the derivative computation method and fusion strategy. revision: yes

Circularity Check

0 steps flagged

No significant circularity: novel architecture and loss are independent proposals

full rationale

The paper proposes replacing discrete residual updates in a transformer encoder with Neural ODE dynamics solved via adaptive integration, adds an auxiliary derivative branch to attention logits, and introduces a customized physics-informed loss as a soft constraint. These components are defined and motivated directly as new design choices rather than being derived from or equivalent to the forecast outputs they produce. Evaluation is performed against external baselines (standard discrete transformer and an existing Neural ODE variant), providing independent comparison. No equations, self-citations, uniqueness theorems, or ansatzes reduce any claimed prediction or result to a fitted input or self-referential definition. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; therefore the ledger is necessarily incomplete. The paper implicitly relies on standard assumptions about Neural ODE solvability and the validity of soft physics constraints as proxies for hard physical laws.

axioms (2)

domain assumption Neural ODEs with adaptive integration can stably replace discrete residual connections in transformer encoders for latent weather dynamics.
Invoked when the paper replaces discrete layer updates with ODE solves.
domain assumption A physics-informed loss term can enforce physical consistency without explicit hard constraints or projection steps.
Central to the training objective described in the abstract.

pith-pipeline@v0.9.0 · 5505 in / 1457 out tokens · 46805 ms · 2026-05-07T09:59:30.293564+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 8 canonical work pages · 2 internal anchors

[1]

Andersson, E.: Medium-range forecasts (2022)

2022
[2]

European Centre for Medium Range Weather Forecasts (2020)

Bauer, P., Quintino, T., Wedi, N., Bonanni, A., Chrust, M., Deconinck, W., Dia- mantakis, M., Düben, P., English, S., Flemming, J., et al.: The ECMWF scalabil- ity programme: Progress and plans. European Centre for Medium Range Weather Forecasts (2020)

2020
[3]

Pangu- weather: A 3d high-resolution model for fast and accurate global weather forecast,

Bi, K., Xie, L., Zhang, H., Chen, X., Gu, X., Tian, Q.: Pangu-weather: A 3d high-resolution model for fast and accurate global weather forecast. arXiv preprint arXiv:2211.02556 (2022)

work page arXiv 2022
[4]

Nature pp

Bi, K., Xie, L., Zhang, H., Chen, X., Gu, X., Tian, Q.: Accurate medium-range global weather forecasting with 3d neural networks. Nature pp. 1–6 (2023)

2023
[5]

Fengwu: Pushing the skillful global medium-range weather forecast beyond 10 days lead,

Chen, K., Han, T., Gong, J., Bai, L., Ling, F., Luo, J.J., Chen, X., Ma, L., Zhang, T., Su, R., et al.: Fengwu: Pushing the skillful global medium-range weather fore- cast beyond 10 days lead. arXiv preprint arXiv:2304.02948 (2023)

work page arXiv 2023
[6]

Chen, L., Zhong, X., Zhang, F., Cheng, Y., Xu, Y., Qi, Y., Li, H.: Fuxi: a cascade machinelearningforecastingsystemfor15-dayglobalweatherforecast.npjClimate and Atmospheric Science6(1), 190 (2023)

2023
[7]

Advances in neural information processing systems31(2018)

Chen, R.T., Rubanova, Y., Bettencourt, J., Duvenaud, D.K.: Neural ordinary dif- ferential equations. Advances in neural information processing systems31(2018)

2018
[8]

arXiv preprint arXiv:2405.14527 (2024)

Couairon, G., Lessig, C., Charantonis, A., Monteleoni, C.: Archesweather: An efficient ai weather forecasting model at 1.5{\deg}resolution. arXiv preprint arXiv:2405.14527 (2024)

work page arXiv 2024
[9]

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

Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, 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 (2020)

work page internal anchor Pith review arXiv 2010
[10]

ecmwf (2023)

ECMWF: Ifs documentation cy48r1. ecmwf (2023)

2023
[11]

GitHub3(2019)

Falcon, W.A.: Pytorch lightning. GitHub3(2019)

2019
[12]

copernicus climate change service (c3s) climate data store (cds) (2018) 14 H

Hersbach, H., Bell, B., Berrisford, P., Biavati, G., Horányi, A., Muñoz Sabater, J., Nicolas,J.,Peubey,C.,Radu,R.,Rozum,I.,etal.:Era5hourlydataonsinglelevels from 1959 to present [dataset]. copernicus climate change service (c3s) climate data store (cds) (2018) 14 H. Saleem et al

1959
[13]

Quarterly Journal of the Royal Meteorological Society146(730), 1999–2049 (2020)

Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz-Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., et al.: The era5 global reanalysis. Quarterly Journal of the Royal Meteorological Society146(730), 1999–2049 (2020)

1999
[14]

Journal of Open Research Software5(1), 10–10 (2017)

Hoyer, S., Hamman, J.: xarray: Nd labeled arrays and datasets in python. Journal of Open Research Software5(1), 10–10 (2017)

2017
[15]

arXiv preprint arXiv:2311.07222 (20 24)

Kochkov, D., Yuval, J., Langmore, I., Norgaard, P., Smith, J., Mooers, G., Lottes, J., Rasp, S., Düben, P., Klöwer, M., et al.: Neural general circulation models. arXiv preprint arXiv:2311.07222 (2023)

work page arXiv 2023
[16]

In: Pro- ceedings of the Platform for Advanced Scientific Computing Conference

Kurth, T., Subramanian, S., Harrington, P., Pathak, J., Mardani, M., Hall, D., Miele, A., Kashinath, K., Anandkumar, A.: Fourcastnet: Accelerating global high- resolution weather forecasting using adaptive fourier neural operators. In: Pro- ceedings of the Platform for Advanced Scientific Computing Conference. pp. 1–11 (2023)

2023
[17]

Science p

Lam, R., Sanchez-Gonzalez, A., Willson, M., Wirnsberger, P., Fortunato, M., Alet, F., Ravuri, S., Ewalds, T., Eaton-Rosen, Z., Hu, W., et al.: Learning skillful medium-range global weather forecasting. Science p. eadi2336 (2023)

2023
[18]

In: Proceedings of the 40th International Conference on Machine Learning

Nguyen, T., Brandstetter, J., Kapoor, A., Gupta, J.K., Grover, A.: Climax: A foundation model for weather and climate. arXiv preprint arXiv:2301.10343 (2023)

work page arXiv 2023
[19]

arXiv preprint arXiv:2312.03876 (2023)

Nguyen, T., Shah, R., Bansal, H., Arcomano, T., Madireddy, S., Maulik, R., Kota- marthi, V., Foster, I., Grover, A.: Scaling transformer neural networks for skillful and reliable medium-range weather forecasting. arXiv preprint arXiv:2312.03876 (2023)

work page arXiv 2023
[20]

Palmer, T., Shutts, G., Hagedorn, R., Doblas-Reyes, F., Jung, T., Leutbecher, M.: Representing model uncertainty in weather and climate prediction. Annu. Rev. Earth Planet. Sci.33, 163–193 (2005)

2005
[21]

Advances in neural information processing sys- tems32(2019)

Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al.: Pytorch: An imperative style, high- performance deep learning library. Advances in neural information processing sys- tems32(2019)

2019
[22]

FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

Pathak, J., Subramanian, S., Harrington, P., Raja, S., Chattopadhyay, A., Mar- dani, M., Kurth, T., Hall, D., Li, Z., Azizzadenesheli, K., et al.: Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural op- erators. arXiv preprint arXiv:2202.11214 (2022)

work page internal anchor Pith review arXiv 2022
[23]

Journal of Ad- vances in Modeling Earth Systems12(11), e2020MS002203 (2020)

Rasp, S., Dueben, P.D., Scher, S., Weyn, J.A., Mouatadid, S., Thuerey, N.: Weath- erbench: a benchmark data set for data-driven weather forecasting. Journal of Ad- vances in Modeling Earth Systems12(11), e2020MS002203 (2020)

2020
[24]

In: The Twelfth International Conference on Learning Rep- resentations (2023)

Verma, Y., Heinonen, M., Garg, V.: Climode: Climate forecasting with physics- informed neural odes. In: The Twelfth International Conference on Learning Rep- resentations (2023)

2023