Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-16T00:16:22.821206Z
Paper Citation Record · LEDGER
As of 22 August 2026, this Paper Citation Record lists 49 of 49 outbound references and 0 inbound Pith citation observations for arXiv:2608.12286.
A citation records a reference. It does not transfer a finding from one paper to another.
Typed states for the displayed outbound observations.
Source: paper_references, paper_reference_links, observed 2026-08-16T00:16:22.821206Z
One-hop event checks from named stored sources.
Source: scholarly_work_events, retraction_status_cache, observed 2026-08-22T06:32:14.747728+00:00
Pith citing papers itemized under the disclosed page cap.
Source: paper_references, paper_reference_links
A source-named dated measurement, never combined with another source.
Source: cited_works
49 of 49 outbound references displayed
External citation measurements
No source-named external measurement is stored.
Observation a1dad3b7-bc47-4f93-95cf-0a899893f1d1 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 1
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 3d64623f-36cb-488d-8bb4-2d26cd1d1223 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Gray and S
Reference 2
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation accf33e8-7216-4511-9a64-6b6ca6a553df · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 3
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 6079e29a-7a4c-4a1c-b2a7-b799e6a69305 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 4
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation afd5eb55-3ae4-42f8-b24a-2d544fae6d9c · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Gierer and H
Reference 5
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 972b747f-1c59-4548-b7e2-b1b9c8175e94 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Solution landscape of reaction-diffusion systems reveals a nonlinear mechanism and spatial robustness of pattern formation.Fundamental Research, 2025
Reference 6
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 9261e871-9dd5-4137-af17-71095b24457d · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Spatial pattern formation in reaction–diffusion models: a computational approach
Reference 7
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation fe4400a3-64ae-45c9-80b9-89ac02e5ba91 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Matrix-oriented discretization methods for reaction–diffusion PDEs: comparisons and applications.Computers and Mathematics with Applications, 79: 2067–2085, 2020
Reference 8
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 179a2e97-2db0-4e86-9bd9-1651734fabf9 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 9
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation b837ec4c-9009-4059-9d7a-4947aa7a5248 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems A homotopy method based on weno schemes for solving steady state problems of hyperbolic conservation laws
Reference 10
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 026426ee-5a87-4315-8a41-02210c308677 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems A bootstrapping approach for computing multiple solutions of differential equations.Journal of Computational and Applied Mathematics, 258:181–190, 2014
Reference 11
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation bb04ad5b-6224-4112-addc-12d8d66d15fe · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 12
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation d35e687a-d584-482d-a506-042aceea4693 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 13
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation b4aa90d7-db19-4776-881e-476c78a4b82e · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 14
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 7bb89543-4a1e-496b-8156-9dd83089bd1b · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 15
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 751d5f72-8c05-48e1-8adf-8b7d5217bf75 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 16
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation bb11d9c2-1e09-44d9-8061-50652efee23b · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Raissi, P
Reference 17
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 7c9f7e68-66cf-43b0-8da5-8ded96c8b7c8 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Giampaolo, M
Reference 18
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation cb196e2c-e105-422b-89e7-f8a5804be879 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 19
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation a8384aff-f089-4f94-af12-d44c66527cb3 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems When and why PINNs fail to train: a neural tangent kernel perspective.Journal of Computational Physics, 449:110768, 2022
Reference 20
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation effcb032-e050-44af-86fa-d23ededc3c2b · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Stiff-PINN: physics-informed neural network for stiff chemical kinetics.The Journal of Physical Chemistry A, 125(36):8098–8106, 2021
Reference 21
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 895763ce-b520-491c-bb75-6f7c8e9b7dc4 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 22
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 030c2407-9a58-41ac-bf10-ce5cef5835b8 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Geometry-informed neural networks.Proceedings of the International Conference on Machine Learning (ICML), 2025
Reference 23
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation cc198c5a-121d-429d-9dce-5cb7cb7686e4 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Newton informed neural operator for computing multiple solutions of nonlinear partial differential equations.Advances in Neural Information Processing Systems (NeurIPS), 2024
Reference 24
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 4ea15d86-9b66-4f22-a090-dd1fc3476ff0 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems A neural network kernel decomposition for learning multiple steady states in parameterized dynamical systems
Reference 25
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 418c771f-ebcd-4a01-b845-c7672cd4d2b2 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Wang and W
Reference 26
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 0f1d4ff5-0f6c-4c76-8ddd-f796b635b045 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 27
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 011aa354-86dc-4b56-98fe-4bf29b5de8f4 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 28
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 1151404d-42d6-42fb-8563-56a3bdec7ab0 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Andersson, Andrew El-Kadi, Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, Remi Lam, and Matthew Willson
Reference 29
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 46f5e0f1-db05-4da2-a12c-051f6c0467b0 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Pawan Kumar, Emilien Dupont, Francisco J
Reference 30
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation cd803664-15ff-475d-a6b2-73102fe31887 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 31
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 74f4efe5-1285-4c56-95c6-09b9607108fd · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Fine-Tune Language Models as Multi-Modal Differential Equation Solvers
Reference 32
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation b87a7d12-23ec-4cbb-abe0-1b261f3abdca · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems In-Context Learning of Linear Systems: Generalization Theory and Applications to Operator Learning
Reference 33
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation 712d853d-949b-46ca-a55f-3fdc0df74798 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unsupervised solution operator learning for mean-field games via sampling- invariant parametrizations.Journal of Computational Physics, 2025
Reference 34
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation f319262c-577e-4876-be51-049cc560c5a8 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Wang, Pierre Erbacher, and Patrick Gallinari
Reference 35
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 003aec4f-cad1-475e-a68c-2fda7909a42c · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems ENMA: tokenwise autoregression for generative neural PDE operators.Preprint at arXiv:2506.06158, 2025
Reference 36
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 8e8474aa-7741-4b4b-8e18-5e8179b68bf9 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Pretrained transformers as universal computation engines.AAAI Conference on Artificial Intelligence, 2022
Reference 37
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 8d16c6ed-ec95-4547-95a1-cf6b32026147 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Efficient approximation of deep ReLU networks for functions on low dimensional manifolds
Reference 38
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 6e2982a1-d1b4-4535-bbe3-bf7a5f6f9636 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Towards a foundation model for partial differential equations: Multi-operator learning and extrapolation.Physical Review E, 111(3):035304, 2025
Reference 39
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation fbf3549c-0e9e-4b2d-981d-fc82cf369473 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Unresolved cited work
Reference 40
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 27e78fa4-609c-4d1f-9305-7140701129e5 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Qwen2.5 Technical Report
Reference 41
Source-reported events for the cited work
Unavailable: canonical work link unavailable.
Observation c673813f-3d29-44bb-94d5-a3abbbe62d8a · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems An efficient quasi-Newton method with tensor product imple- mentation for solving quasi-linear elliptic equations and systems.Journal of Scientific Computing, 103:89, 2025
Reference 42
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 99a2878c-0385-4510-a204-fa592a149e88 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Stepien, Erica M
Reference 43
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 7e23df48-dfa8-4299-b612-b83eb1403cb0 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Aronow, and Leili Shahri- yari
Reference 44
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 235b21be-aa01-49d5-ae17-6a6f0daeda46 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Rempala, and Hye-Won Kang
Reference 45
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 6af54da7-c7bb-49a2-8ec4-7f76e63289cd · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems A topology-based network tree for the prediction of protein– protein binding affinity changes following mutation.Nature Machine Intelligence, 2(2):116–123, 2020
Reference 46
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 8180b5ed-0bb3-4f81-8f0b-820e91b128a3 · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Order matters: sequence to sequence for sets.International Conference on Learning Representations (ICLR), 2016
Reference 47
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation 100d874e-aae7-4d0f-bc6b-9f5b76d3780f · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems End-to-end object detection with transformers.European Conference on Computer Vision (ECCV), 2020
Reference 48
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
Observation cd917e7a-71a5-49bb-b9eb-3667b0674c3b · outbound
PatternFormer: Learning Multiple Solution Patterns in Reaction--Diffusion Systems Fleet, and Geoffrey Hinton
Reference 49
Source-reported events for the cited work
No event found in the named queried sources as of 2026-08-22T06:32:14.747728+00:00.
No inbound Pith citation observations are available.