arxiv: 2602.16213 · v2 · submitted 2026-02-18 · 💻 cs.LG · cs.AI· cs.CV· physics.comp-ph

Recognition: 2 theorem links

· Lean Theorem

Graph neural network for colliding particles with an application to sea ice floe modeling

Ruibiao Zhu

Authors on Pith no claims yet

Pith reviewed 2026-05-15 21:24 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.CVphysics.comp-ph

keywords graph neural networksea ice modelingcollision capturedata assimilationtrajectory predictionone-dimensional simulationmarginal ice zone

0 comments

The pith

A graph neural network simulates one-dimensional sea ice floe collisions as accurately as traditional methods but much faster.

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

The paper presents a graph neural network approach for modeling sea ice floes by representing individual ice pieces as nodes and their collisions as edges in a one-dimensional framework. This Collision-captured Network incorporates data assimilation to learn the dynamics from synthetic data. It shows that the model can predict trajectories under various conditions without observed data or with some data points available. The result is faster simulation of the ice movements while preserving the accuracy of conventional numerical techniques. Such efficiency could improve forecasting for sea ice in areas where it meets open water.

Core claim

The authors introduce the Collision-captured Network, a graph neural network model that uses nodes to represent sea ice pieces and edges to capture physical interactions and collisions. Combined with data assimilation methods, the network learns to simulate the trajectories of these pieces in one dimension. Validation on synthetic data demonstrates that the approach accelerates the computation of ice dynamics compared to standard numerical solvers while maintaining equivalent accuracy, whether or not additional observation data is provided.

What carries the argument

The graph representation of sea ice where nodes correspond to individual floes and edges encode collision and interaction forces, serving as the input structure for the Collision-captured Network that learns the time evolution via graph neural network layers integrated with data assimilation.

If this is right

Simulations of sea ice trajectories become computationally cheaper without loss of accuracy.
The model handles cases with missing observation data effectively.
It offers a scalable tool for predictions in marginal ice zones.
Machine learning combined with data assimilation can replace intensive numerical integration for particle systems.

Where Pith is reading between the lines

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

This one-dimensional success suggests the graph approach could extend to two or three dimensions for more realistic sea ice modeling.
Similar graph structures might help simulate other colliding particle systems like dust grains or granular materials.
Integrating this with real satellite data could lead to operational forecasting systems for ice-covered oceans.

Load-bearing premise

The graph with ice pieces as nodes and collisions as edges is sufficient to represent the dominant interactions in one-dimensional sea ice motion.

What would settle it

A test case where the graph neural network predictions for floe positions and velocities differ substantially from those produced by a high-fidelity numerical collision simulator on the same initial conditions.

read the original abstract

This paper introduces a novel approach to sea ice modeling using Graph Neural Networks (GNNs), utilizing the natural graph structure of sea ice, where nodes represent individual ice pieces, and edges model the physical interactions, including collisions. This concept is developed within a one-dimensional framework as a foundational step. Traditional numerical methods, while effective, are computationally intensive and less scalable. By utilizing GNNs, the proposed model, termed the Collision-captured Network (CN), integrates data assimilation (DA) techniques to effectively learn and predict sea ice dynamics under various conditions. The approach was validated using synthetic data, both with and without observed data points, and it was found that the model accelerates the simulation of trajectories without compromising accuracy. This advancement offers a more efficient tool for forecasting in marginal ice zones (MIZ) and highlights the potential of combining machine learning with data assimilation for more effective and efficient modeling.

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

This applies standard GNN message passing to 1D sea ice floe collisions with data assimilation, but the abstract supplies no error metrics or baselines to support the no-accuracy-loss claim.

read the letter

The core move is taking existing graph neural networks for particle interactions and mapping them onto one-dimensional sea ice floes, with nodes as individual pieces and edges as collisions. They add data assimilation and test on synthetic trajectories both with and without observations, claiming the result runs faster than traditional integration while keeping accuracy intact. That framing is reasonable as a first step toward faster surrogates for marginal ice zone forecasts.

Referee Report

2 major / 2 minor

Summary. The paper introduces a Collision-captured Network (CN), a graph neural network for modeling colliding particles with application to one-dimensional sea ice floe dynamics. Nodes represent individual ice pieces and edges model physical interactions including collisions. The model integrates data assimilation techniques, is trained and validated on synthetic data both with and without observations, and claims to accelerate trajectory simulations while preserving accuracy compared to traditional numerical methods.

Significance. If the central claim holds with proper quantitative support, the work could provide a scalable machine-learning alternative to computationally intensive traditional methods for sea ice modeling in marginal ice zones, by exploiting the natural graph structure of floes and combining GNN message passing with data assimilation.

major comments (2)

[Abstract] Abstract: the central claim that accuracy is preserved is unsupported by any quantitative error metrics, baseline comparisons (e.g., against explicit Euler or event-driven integrators), or description of how accuracy was measured (e.g., position RMSE, momentum conservation error), preventing verification that the result holds.
[Method] Method section on graph construction: proximity-based or learned edges with standard message passing do not explicitly enforce sequential impulse resolution or local momentum conservation required for 1D multi-body collisions; this risks averaging impulses across neighbors and allowing penetration or long-term drift, and the synthetic-data validation does not test dense or long-horizon regimes where this issue would be load-bearing.

minor comments (2)

[Abstract] Abstract: the acronym 'CN' for Collision-captured Network is introduced without prior expansion or definition.
[Introduction] The one-dimensional setting is presented as a foundational step, but no discussion is given of how the approach would extend to 2D floe interactions or what additional graph features would be required.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that the manuscript requires stronger quantitative support for the accuracy claims and clarification on physical constraint enforcement. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that accuracy is preserved is unsupported by any quantitative error metrics, baseline comparisons (e.g., against explicit Euler or event-driven integrators), or description of how accuracy was measured (e.g., position RMSE, momentum conservation error), preventing verification that the result holds.

Authors: We acknowledge the absence of explicit quantitative metrics in the abstract and main text. In the revised manuscript we will add position RMSE, velocity RMSE, and total momentum conservation error (L2 norm of momentum drift) for the Collision-captured Network against ground-truth synthetic trajectories. We will also include direct comparisons to explicit Euler and event-driven integrators on the same test sets, reporting both accuracy and wall-clock speedup. Accuracy measurement will be described as mean trajectory error over 1000-step rollouts on held-out synthetic sequences with and without assimilated observations. revision: yes
Referee: [Method] Method section on graph construction: proximity-based or learned edges with standard message passing do not explicitly enforce sequential impulse resolution or local momentum conservation required for 1D multi-body collisions; this risks averaging impulses across neighbors and allowing penetration or long-term drift, and the synthetic-data validation does not test dense or long-horizon regimes where this issue would be load-bearing.

Authors: The concern about impulse resolution and conservation is well taken. Our current message-passing layer learns a collision operator that is trained to conserve pairwise momentum, yet it does not explicitly order impulses. We will revise the method to insert a 1D sequential impulse-resolution step (sorting nodes by position before message passing) and will add a hard constraint layer that projects velocities to enforce local momentum conservation after each update. We will also extend the experimental section with new synthetic tests on dense packings (inter-floe gaps < 0.1) and long-horizon rollouts (10,000 steps) to quantify penetration events and cumulative drift, reporting these metrics alongside the existing results. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical GNN training on synthetic data for 1D floe trajectories

full rationale

The paper presents an empirical Collision-captured Network (CN) that trains a GNN on synthetic sea-ice trajectory data in one dimension, with nodes as floes and edges as collisions, then integrates data assimilation for prediction. No derivation chain, equations, or self-citations are shown that reduce the claimed acceleration of simulations to a fitted parameter by construction, a self-defined quantity, or a load-bearing uniqueness theorem from the same authors. Validation on held-out synthetic data (with and without observations) supplies independent empirical support rather than tautological equivalence, making the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The abstract provides no explicit free parameters, axioms, or invented entities; the modeling rests on the unstated assumption that standard GNN layers plus data assimilation suffice for the collision dynamics.

pith-pipeline@v0.9.0 · 5454 in / 1067 out tokens · 35224 ms · 2026-05-15T21:24:32.177068+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.

IndisputableMonolith/Foundation/Atomicity.lean atomic_tick, sequential_preserves_conservation echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The proposed Collision-captured Network (CN) … Ef = ϕ_θ1([R_r^T X_{t-2:t-1}; R_s^T X_{t-2:t-1}; E_{t-1}]), X_t = γ_θ2([X_{t-2:t-1}; R_r E_f]) … Mish activation … trained on synthetic DEM data
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

no mention of cost functions, golden ratio, recognition ladder or 8-tick structure

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

Works this paper leans on

91 extracted references · 91 canonical work pages · 3 internal anchors

[1]

Nature Climate Change 8(7), 599–603 (2018) https://doi.org/10

Massonnet, F., Vancoppenolle, M., Goosse, H., Docquier, D., Fichefet, T., Blanchard- Wrigglesworth, E.: Arctic sea-ice change tied to its mean state through thermody- namic processes. Nature Climate Change 8(7), 599–603 (2018) https://doi.org/10. 1038/s41558-018-0204-z

work page 2018
[2]

The Cryosphere 12(10), 3373–3382 (2018) https://doi.org/10

Letterly, A., Key, J., Liu, Y.: Arctic cli- mate: changes in sea ice extent outweigh changes in snow cover. The Cryosphere 12(10), 3373–3382 (2018) https://doi.org/10. 5194/tc-12-3373-2018

work page 2018
[3]

Nature communications11(1), 5177 (2020)

Wunderling, N., Willeit, M., Donges, J.F., Winkelmann, R.: Global warming due to loss of large ice masses and arctic summer sea ice. Nature communications11(1), 5177 (2020)

work page 2020
[4]

Science354(6313), 747–750 (2016) https://doi.org/10.1126/science.aag2345

Notz, D., Stroeve, J.: Observed arctic sea- ice loss directly follows anthropogenic co2 emission. Science354(6313), 747–750 (2016) https://doi.org/10.1126/science.aag2345

work page doi:10.1126/science.aag2345 2016
[5]

Blockley, E., Vancoppenolle, M., Hunke, E., Bitz, C., Feltham, D., Lemieux, J.-F., Losch, M., Maisonnave, E., Notz, D., Ram- pal, P., Tietsche, S., Tremblay, B., Turner, A., Massonnet, F., ´Olason, E., Roberts, A., Aksenov, Y., Fichefet, T., Garric, G., Iovino, D., Madec, G., Rousset, C., Melia, D.S., Schroeder, D.: The future of sea ice model- ing: Where...

work page 2020
[6]

Nature Climate Change9(2), 123–129 (2019) https://doi

Mori, M., Kosaka, Y., Watanabe, M., Naka- mura, H., Kimoto, M.: A reconciled esti- mate of the influence of Arctic sea-ice loss on recent Eurasian cooling. Nature Climate Change9(2), 123–129 (2019) https://doi. org/10.1038/s41558-018-0379-3

work page doi:10.1038/s41558-018-0379-3 2019
[7]

Geophysical Research Letters 45(7), 3255–3263 (2018) https://doi.org/10

Ogawa, F., Keenlyside, N., Gao, Y.,et al.: Evaluating impacts of recent arctic sea ice loss on the northern hemisphere winter cli- mate change. Geophysical Research Letters 45(7), 3255–3263 (2018) https://doi.org/10. 1002/2017GL076502

work page 2018
[8]

Geophysical Research Letters41(4), 1216–1225 (2014) https://doi.org/10.1002/ 2013GL058951

Stroeve, J.C., Markus, T., Boisvert, L., Miller, J., Barrett, A.: Changes in arctic melt season and implications for sea ice loss. Geophysical Research Letters41(4), 1216–1225 (2014) https://doi.org/10.1002/ 2013GL058951

work page 2014
[9]

Nature Ecology & Evolution, 1–6 (2024)

Cunningham, C.X., Williamson, G.J., Bow- man, D.M.: Increasing frequency and inten- sity of the most extreme wildfires on earth. Nature Ecology & Evolution, 1–6 (2024)

work page 2024
[10]

Ambio 46(Suppl 3), 341–354 (2017) https://doi.org/ 10.1007/s13280-017-0953-3

Cr´ epin, A.S., Karcher, M., Gascard, J.C.: Arctic climate change, economy and soci- ety (access): Integrated perspectives. Ambio 46(Suppl 3), 341–354 (2017) https://doi.org/ 10.1007/s13280-017-0953-3

work page doi:10.1007/s13280-017-0953-3 2017
[11]

Ambio 49(2), 407–418 (2020) https://doi.org/10

Alvarez, J., Yumashev, D., Whiteman, 14 G.: A framework for assessing the eco- nomic impacts of arctic change. Ambio 49(2), 407–418 (2020) https://doi.org/10. 1007/s13280-019-01211-z

work page 2020
[12]

GLOBAL NEST JOURNAL27(6) (2025) https://doi.org/10.30955/gnj.07421

Hou, Z., Liu, J., Pang, J.: Green transition and climate adaptation: empirical evidence of corporate resilience in the low-carbon economy. GLOBAL NEST JOURNAL27(6) (2025) https://doi.org/10.30955/gnj.07421

work page doi:10.30955/gnj.07421 2025
[13]

Marine Pollution Bulletin 166, 112223 (2021) https://doi.org/10.1016/ j.marpolbul.2021.112223

Ani, C.J., Robson, B.: Responses of marine ecosystems to climate change impacts and their treatment in biogeochemical ecosys- tem models. Marine Pollution Bulletin 166, 112223 (2021) https://doi.org/10.1016/ j.marpolbul.2021.112223

work page arXiv 2021
[14]

Nature Climate Change, 1–10 (2024)

Lubitz, N., Daly, R., Smoothey, A.F., Vianello, P., Roberts, M.J., Schoeman, D.S., Sheaves, M., Cowley, P.D., Dagorn, L., For- get, F.G., et al.: Climate change-driven cool- ing can kill marine megafauna at their distri- butional limits. Nature Climate Change, 1–10 (2024)

work page 2024
[15]

Nature Climate Change, 1–4 (2024)

Tollenaar, V., Zekollari, H., Kittel, C., Farinotti, D., Lhermitte, S., Debaille, V., Goderis, S., Claeys, P., Joy, K.H., Pattyn, F.: Antarctic meteorites threatened by cli- mate warming. Nature Climate Change, 1–4 (2024)

work page 2024
[16]

Nature 619(7968), 102–111 (2023) https://doi.org/ 10.1038/s41586-023-06083-8

Rockstr¨ om, J., Gupta, J., Qin, D.,et al.: Safe and just earth system boundaries. Nature 619(7968), 102–111 (2023) https://doi.org/ 10.1038/s41586-023-06083-8

work page doi:10.1038/s41586-023-06083-8 2023
[17]

Xu, Z., Tartakovsky, A.M., Pan, W.: Discrete-element model for the interaction between ocean waves and sea ice. Phys. Rev. E85, 016703 (2012) https://doi.org/10. 1103/PhysRevE.85.016703

work page 2012
[18]

Geoscientific Model Development9(3), 1219–1241 (2016)

Herman, A.: Discrete-element bonded- particle sea ice model design, version 1.3 a–model description and implementation. Geoscientific Model Development9(3), 1219–1241 (2016)

work page 2016
[19]

Jour- nal of Advances in Modeling Earth Sys- tems14(6) (2022) https://doi.org/10.1029/ 2021MS002614

West, B., O’Connor, D., Parno, M., Krackow, M., Polashenski, C.: Bonded discrete ele- ment simulations of sea ice with non-local failure: Applications to nares strait. Jour- nal of Advances in Modeling Earth Sys- tems14(6) (2022) https://doi.org/10.1029/ 2021MS002614

work page 2022
[20]

SIAM Multiscale Modeling and Simulation (MMS) (2023)

Deng, Q., Stechmann, S.N., Cheng, N.: Particle-continuum multiscale modeling of sea ice floes. SIAM Multiscale Modeling and Simulation (MMS) (2023)

work page 2023
[21]

Journal of Open Source Software8(88), 5039 (2023) https://doi.org/10.21105/joss.05039

Montemuro, B.P., Manucharyan, G.E.: Sub- zero: a discrete element sea ice model that simulates floes as evolving concave polygons. Journal of Open Source Software8(88), 5039 (2023) https://doi.org/10.21105/joss.05039

work page doi:10.21105/joss.05039 2023
[22]

Engineering Computations 21(2/3/4), 409–421 (2004) https://doi.org/ 10.1108/02644400410519857

Hopkins, M.A.: A discrete element lagrangian sea ice model. Engineering Computations 21(2/3/4), 409–421 (2004) https://doi.org/ 10.1108/02644400410519857

work page doi:10.1108/02644400410519857 2004
[23]

Geoscientific Model Devel- opment15(5), 1953–1970 (2022) https://doi

Turner, A.K., Peterson, K.J., Bolintineanu, D.: Geometric remapping of particle distri- butions in the discrete element model for sea ice (demsi v0.0). Geoscientific Model Devel- opment15(5), 1953–1970 (2022) https://doi. org/10.5194/gmd-15-1953-2022

work page doi:10.5194/gmd-15-1953-2022 1953
[24]

SIAM, Philadelphia (2005)

Tarantola, A.: Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM, Philadelphia (2005). https://doi.org/ 10.1137/1.9780898717921

work page doi:10.1137/1.9780898717921 2005
[25]

Springer, Philadelphia (2018)

Isakov, V.: Inverse Problems for Partial Dif- ferential Equations. Springer, Philadelphia (2018)

work page 2018
[26]

In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining

Ying, R., He, R., Chen, K., Eksombatchai, P., Hamilton, W.L., Leskovec, J.: Graph convo- lutional neural networks for web-scale recom- mender systems. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. KDD ’18, pp. 974–983. Association for Comput- ing Machinery, New York, NY, USA (2018). https://doi.org/1...

work page doi:10.1145/3219819.3219890 2018
[27]

Nature communications 12(1), 3168 (2021)

Gligorijevi´ c, V., Renfrew, P.D., Kosciolek, T., Leman, J.K., Berenberg, D., Vatanen, T., Chandler, C., Taylor, B.C., Fisk, I.M., 15 Vlamakis, H.,et al.: Structure-based pro- tein function prediction using graph con- volutional networks. Nature communications 12(1), 3168 (2021)

work page 2021
[28]

Nature 624(7990), 80–85 (2023)

Merchant, A., Batzner, S., Schoenholz, S.S., Aykol, M., Cheon, G., Cubuk, E.D.: Scaling deep learning for materials discovery. Nature 624(7990), 80–85 (2023)

work page 2023
[29]

IEEE transactions on neural networks and learning systems32(1), 4–24 (2020)

Wu, Z., Pan, S., Chen, F., Long, G., Zhang, C., Philip, S.Y.: A comprehensive survey on graph neural networks. IEEE transactions on neural networks and learning systems32(1), 4–24 (2020)

work page 2020
[30]

AI open1, 57–81 (2020)

Zhou, J., Cui, G., Hu, S., Zhang, Z., Yang, C., Liu, Z., Wang, L., Li, C., Sun, M.: Graph neural networks: A review of methods and applications. AI open1, 57–81 (2020)

work page 2020
[31]

Mathematics 13(11), 1785 (2025) https://doi.org/10.3390/ math13111785

Liu, J., Hou, Z., Liu, B., Zhou, X.: Math- ematical and machine learning innovations for power systems: Predicting transformer oil temperature with beluga whale optimization- based hybrid neural networks. Mathematics 13(11), 1785 (2025) https://doi.org/10.3390/ math13111785

work page 2025
[32]

Electronics14(12), 2329 (2025) https://doi

Hou, Z., Liu, J., Shao, Z., Ma, Q., Liu, W.: Machine learning innovations in renewable energy systems with integrated nrbo-txad for enhanced wind speed forecasting accuracy. Electronics14(12), 2329 (2025) https://doi. org/10.3390/electronics14122329

work page doi:10.3390/electronics14122329 2025
[33]

Heliyon10(18) (2024) https://doi.org/10

Xie, J., He, J., Gao, Z., Wang, S., Liu, J., Fan, H.: An enhanced snow abla- tion optimizer for uav swarm path plan- ning and engineering design problems. Heliyon10(18) (2024) https://doi.org/10. 1016/j.heliyon.2024.e37819

work page 2024
[34]

Entropy26(5), 402 (2024) https://doi.org/10.3390/e26050402

Liu, J., Duan, Z., Hu, X., Zhong, J., Yin, Y.: Detracking autoencoding conditional genera- tive adversarial network: Improved generative adversarial network method for tabular miss- ing value imputation. Entropy26(5), 402 (2024) https://doi.org/10.3390/e26050402

work page doi:10.3390/e26050402 2024
[35]

Advances in neural information processing systems29(2016)

Battaglia, P., Pascanu, R., Lai, M., Jimenez Rezende, D., et al.: Interaction net- works for learning about objects, relations and physics. Advances in neural information processing systems29(2016)

work page 2016
[36]

Journal of Geophysical Research: Oceans104(C7), 15815–15825 (1999) https://doi.org/10.1029/ 1999JC900127

Hopkins, M.A., Tuhkuri, J.: Compres- sion of floating ice fields. Journal of Geophysical Research: Oceans104(C7), 15815–15825 (1999) https://doi.org/10.1029/ 1999JC900127

work page 1999
[37]

Cold Regions Science and Tech- nology78, 31–39 (2012) https://doi.org/10

Sun, S., Shen, H.H.: Simulation of pancake ice load on a circular cylinder in a wave and current field. Cold Regions Science and Tech- nology78, 31–39 (2012) https://doi.org/10. 1016/j.coldregions.2012.02.003

work page 2012
[38]

Journal of Advances in Modeling Earth Systems10(9), 2228–2244 (2018) https://doi.org/10.1029/ 2018MS001299

Damsgaard, A., Adcroft, A., Sergienko, O.: Application of discrete element methods to approximate sea ice dynamics. Journal of Advances in Modeling Earth Systems10(9), 2228–2244 (2018) https://doi.org/10.1029/ 2018MS001299

work page 2018
[39]

Computers and Geotechnics 49, 299–313 (2013) https://doi.org/10.1016/ j.compgeo.2012.09.001

Obermayr, M., Dressler, K., Vrettos, C., Eberhard, P.: A bonded-particle model for cemented sand. Computers and Geotechnics 49, 299–313 (2013) https://doi.org/10.1016/ j.compgeo.2012.09.001

work page 2013
[40]

Granu- lar Matter13(4), 341–364 (2011) https://doi

Ergenzinger, C., Seifried, R., Eberhard, P.: A discrete element model to describe failure of strong rock in uniaxial compression. Granu- lar Matter13(4), 341–364 (2011) https://doi. org/10.1007/s10035-010-0230-7

work page doi:10.1007/s10035-010-0230-7 2011
[41]

G´ eotechnique29(1), 47–65 (1979) https:// doi.org/10.1680/geot.1979.29.1.47

Cundall, P.A., Strack, O.D.L.: A discrete numerical model for granular assemblies. G´ eotechnique29(1), 47–65 (1979) https:// doi.org/10.1680/geot.1979.29.1.47

work page doi:10.1680/geot.1979.29.1.47 1979
[42]

Geophysical Research Letters 43(23), 12165–12173 (2016) https://doi.org/ 10.1002/2016GL071579

Damsgaard, A., Egholm, D.L., Beem, L.H., Tulaczyk, S., Larsen, N.K., Piotrowski, J.A., Siegfried, M.R.: Ice flow dynamics forced by water pressure variations in subglacial granular beds. Geophysical Research Letters 43(23), 12165–12173 (2016) https://doi.org/ 10.1002/2016GL071579

work page doi:10.1002/2016gl071579 2016
[43]

Granular 16 Matter13(4), 341–364 (2011)

Ergenzinger, C., Seifried, R., Eberhard, P.: A discrete element model to describe failure of strong rock in uniaxial compression. Granular 16 Matter13(4), 341–364 (2011)

work page 2011
[44]

European journal of environmental and civil engineering12(7-8), 785–826 (2008)

Luding, S.: Introduction to discrete element methods: basic of contact force models and how to perform the micro-macro transition to continuum theory. European journal of environmental and civil engineering12(7-8), 785–826 (2008)

work page 2008
[45]

Journal of Advances in Modeling Earth Systems13(7), 2020–002336 (2021)

Damsgaard, A., Sergienko, O., Adcroft, A.: The effects of ice floe-floe interactions on pressure ridging in sea ice. Journal of Advances in Modeling Earth Systems13(7), 2020–002336 (2021)

work page 2020
[46]

Journal of Geophysical Research: Oceans123(6), 4298–4321 (2018)

Lund, B., Graber, H.C., Persson, P., Smith, M., Doble, M., Thomson, J., Wadhams, P.: Arctic sea ice drift measured by ship- board marine radar. Journal of Geophysical Research: Oceans123(6), 4298–4321 (2018)

work page 2018
[47]

Hanley, K., O’Sullivan, C.: Analytical study of the accuracy of discrete element simulations

J. Hanley, K., O’Sullivan, C.: Analytical study of the accuracy of discrete element simulations. International Journal for Numer- ical Methods in Engineering109(1), 29–51 (2017)

work page 2017
[48]

Engineering Computations 21, 278–303 (2004) https://doi.org/10.1108/ 02644400410519794

O’Sullivan, C., Bray, J.: Selecting a suitable time step for discrete element simulations that use the central difference time inte- gration scheme. Engineering Computations 21, 278–303 (2004) https://doi.org/10.1108/ 02644400410519794

work page 2004
[49]

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences376(2129), 20170335 (2018)

Tuhkuri, J., Poloj¨ arvi, A.: A review of dis- crete element simulation of ice–structure interaction. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences376(2129), 20170335 (2018)

work page 2018
[50]

JOM51(2), 21–27 (1999) https://doi.org/10.1007/s11837-999-0206-4

Schulson, E.M.: The structure and mechani- cal behavior of ice. JOM51(2), 21–27 (1999) https://doi.org/10.1007/s11837-999-0206-4

work page doi:10.1007/s11837-999-0206-4 1999
[51]

Jour- nal of Automated Reasoning26(2), 107–137 (2001) https://doi.org/10.1023/A: 1026518331905

Petrovic, J.J.: Review mechanical properties of ice and snow. Journal of Materials Science 38(1), 1–6 (2003) https://doi.org/10.1023/A: 1021134128038

work page doi:10.1023/a: 2003
[52]

Advances in neural information processing systems32 (2019)

Greydanus, S., Dzamba, M., Yosinski, J.: Hamiltonian neural networks. Advances in neural information processing systems32 (2019)

work page 2019
[53]

In: International Conference on Machine Learning, pp

Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., Battaglia, P.: Learn- ing to simulate complex physics with graph networks. In: International Conference on Machine Learning, pp. 8459–8468 (2020). PMLR

work page 2020
[54]

Cog- nitive Science14(2), 179–211 (1990) https: //doi.org/10.1207/s15516709cog1402 1

Elman, J.L.: Finding structure in time. Cog- nitive Science14(2), 179–211 (1990) https: //doi.org/10.1207/s15516709cog1402 1

work page doi:10.1207/s15516709cog1402 1990
[55]

Journal of Chemical Information and Modeling45(5), 1159–1168 (2005) https://doi.org/10.1021/ ci049613b

Merkwirth, C., Lengauer, T.: Automatic generation of complementary descriptors with molecular graph networks. Journal of Chemical Information and Modeling45(5), 1159–1168 (2005) https://doi.org/10.1021/ ci049613b . PMID: 16180893

work page 2005
[56]

IEEE Transactions on Neural Networks20, 61–80 (2009)

Scarselli, F., Gori, M., Tsoi, A.C., Hagen- buchner, M., Monfardini, G.: The graph neu- ral network model. IEEE Transactions on Neural Networks20, 61–80 (2009)

work page 2009
[57]

BMVC (2020)

Misra, D.: Mish: A self regularized non- monotonic activation function. BMVC (2020)

work page 2020
[58]

nature 405(6789), 947–951 (2000)

Hahnloser, R.H., Sarpeshkar, R., Mahowald, M.A., Douglas, R.J., Seung, H.S.: Digital selection and analogue amplification coex- ist in a cortex-inspired silicon circuit. nature 405(6789), 947–951 (2000)

work page 2000
[59]

ICLR (2015)

Kingma, D.P.: Adam: A method for stochas- tic optimization. ICLR (2015)

work page 2015
[60]

In: International Con- ference on Machine Learning, pp

Pham, H., Guan, M., Zoph, B., Le, Q., Dean, J.: Efficient neural architecture search via parameters sharing. In: International Con- ference on Machine Learning, pp. 4095–4104 (2018). PMLR

work page 2018
[61]

Gaussian Error Linear Units (GELUs)

Hendrycks, D.: Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415 (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
[62]

Searching for Activation Functions

Ramachandran, P., Zoph, B., Le, Q.V.: Searching for activation functions. arXiv preprint arXiv:1710.05941 (2017) 17

work page internal anchor Pith review Pith/arXiv arXiv 2017
[63]

Neural Networks130, 195–205 (2020)

Nikolentzos, G., Dasoulas, G., Vazirgiannis, M.: K-hop graph neural networks. Neural Networks130, 195–205 (2020)

work page 2020
[64]

Advances in Neural Information Processing Systems36, 35084–35106 (2023)

Wu, X., Ajorlou, A., Wu, Z., Jadbabaie, A.: Demystifying oversmoothing in attention- based graph neural networks. Advances in Neural Information Processing Systems36, 35084–35106 (2023)

work page 2023
[65]

Machine Learning114(8), 184 (2025)

Kelesis, D., Fotakis, D., Paliouras, G.: Ana- lyzing the effect of residual connections to oversmoothing in graph neural networks. Machine Learning114(8), 184 (2025)

work page 2025
[66]

Climate Dynamics62(3), 2475–2498 (2024)

Salazar, ´A., Thatcher, M., Goubanova, K., Bernal, P., Guti´ errez, J., Squeo, F.: Cmip6 precipitation and temperature projections for chile. Climate Dynamics62(3), 2475–2498 (2024)

work page 2024
[67]

Quarterly Journal of the Royal Meteo- rological Society150(759), 1048–1067 (2024)

Christophersen, J.A., Rydbeck, A., Flatau, M., Janiga, M., Reynolds, C.A., Jensen, T., Smith, T.: Oceanic rossby wave predictabil- ity in ecmwf’s subseasonal-to-seasonal refore- casts. Quarterly Journal of the Royal Meteo- rological Society150(759), 1048–1067 (2024)

work page 2024
[68]

In: Das, K.N., Bansal, J.C., Deep, K., Nagar, A.K., Pathipooranam, P., Naidu, R.C

Dubey, S.P., Kumar, M.S., Balaji, S.: Gpu computing for compute-intensive scientific calculation. In: Das, K.N., Bansal, J.C., Deep, K., Nagar, A.K., Pathipooranam, P., Naidu, R.C. (eds.) Soft Computing for Prob- lem Solving, pp. 131–140. Springer, Singapore (2020)

work page 2020
[69]

R., Millman, K

Harris, C.R., Millman, K.J., Walt, S.J., Gom- mers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., Kerkwijk, M.H., Brett, M., Haldane, A., R´ ıo, J.F., Wiebe, M., Peterson, P., G´ erard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C., Oliphant, T.E.: Array progr...

work page doi:10.1038/s41586-020-2649-2 2020
[70]

Advances in neural informa- tion processing systems32(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 informa- tion processing systems32(2019)

work page 2019
[71]

In: NIPS 2017 Workshop on Autodiff (2017)

Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., Lerer, A.: Automatic differentiation in pytorch. In: NIPS 2017 Workshop on Autodiff (2017). https://openreview.net/forum?id=BJJsrmfCZ

work page 2017
[72]

Journal of Geophysical Research: Oceans99(C5), 10143–10162 (1994) https: //doi.org/10.1029/94JC00572

Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. Journal of Geophysical Research: Oceans99(C5), 10143–10162 (1994) https: //doi.org/10.1029/94JC00572

work page doi:10.1029/94jc00572 1994
[73]

Monthly Weather Review126(6), 1719–1724 (1998) https://doi.org/10.1175/1520-0493(1998) 126⟨1719:ASITEK⟩2.0.CO;2

Burgers, G., Leeuwen, P.J., Evensen, G.: Analysis scheme in the ensem- ble kalman filter. Monthly Weather Review126(6), 1719–1724 (1998) https://doi.org/10.1175/1520-0493(1998) 126⟨1719:ASITEK⟩2.0.CO;2

work page doi:10.1175/1520-0493(1998 1998
[74]

part i: Theoretical aspects

Bishop, C.H., Etherton, B.J., Majum- dar, S.J.: Adaptive sampling with the ensemble transform kalman filter. part i: Theoretical aspects. Monthly Weather Review129(3), 420–436 (2001) https://doi.org/10.1175/1520-0493(2001) 129⟨0420:ASWTET⟩2.0.CO;2

work page doi:10.1175/1520-0493(2001 2001
[75]

Tellus A: Dynamic Mete- orology and Oceanography59(2), 225–237 (2007)

Harlim, J., Hunt, B.R.: A non-gaussian ensemble filter for assimilating infrequent noisy observations. Tellus A: Dynamic Mete- orology and Oceanography59(2), 225–237 (2007)

work page 2007
[76]

The Cryosphere15(8), 3681–3698 (2021)

Lavergne, T., Pi˜ nol Sol´ e, M., Down, E., Donlon, C.: Towards a swath-to-swath sea- ice drift product for the copernicus imag- ing microwave radiometer mission. The Cryosphere15(8), 3681–3698 (2021)

work page 2021
[77]

Remote Sensing14(5), 1261 (2022)

Wang, X., Chen, R., Li, C., Chen, Z., Hui, F., Cheng, X.: An intercomparison of satel- lite derived arctic sea ice motion products. Remote Sensing14(5), 1261 (2022)

work page 2022
[78]

Zhu, R.: Graph neural network for colliding particles with an application to sea ice floe modeling.figsharehttps://doi.org/10.6084/ 18 m9.figshare.28658180 (2025)

work page 2025
[79]

Ionics31(10), 10643–10669 (2025) https://doi.org/10.1007/ s11581-025-06578-6

Liu, J., Hou, Z., Xu, Y., He, Y., Wang, B.: Enhanced beluga whale optimization meets gru and adaptive cubature kalman filter: a novel approach for state of charge estima- tion in lithium-ion batteries. Ionics31(10), 10643–10669 (2025) https://doi.org/10.1007/ s11581-025-06578-6

work page 2025
[80]

Philosoph- ical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sci- ences371(1987), 20120375 (2013)

Barab´ asi, A.-L.: Network science. Philosoph- ical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sci- ences371(1987), 20120375 (2013)

work page 1987

Showing first 80 references.