{"paper":{"title":"U-HNO: A U-shaped Hybrid Neural Operator with Sparse-Point Adaptive Routing for Non-stationary PDE Dynamics","license":"http://creativecommons.org/licenses/by/4.0/","headline":"U-HNO uses per-point hard masks to route between global Fourier and local Gaussian branches for PDEs with mixed smooth and sharp dynamics.","cross_cats":["cs.NA","math.NA"],"primary_cat":"cs.LG","authors_text":"Jinliang Liu, Xiao Yang, Yingzhe Ma, Yuxin Xie, Zihan Xiong","submitted_at":"2026-05-13T04:00:43Z","abstract_excerpt":"Solutions to many partial differential equations (PDEs) display coexisting smooth global transport and localized sharp features within a single trajectory: shock fronts, thin interfaces, and concentrated high-frequency content sit on top of slowly varying backgrounds. This poses a challenge for neural operators: Fourier-based architectures mix nonlocal interactions efficiently but tend to under-resolve localized non-smooth features, whereas spatially local architectures recover fine detail at the cost of long-range propagation and rollout stability. Existing hybrid operators paper over this te"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that a per-pixel hard mask based on local contrast of the routing signal can reliably select the appropriate branch (global or local) without causing training instabilities or degrading performance on smooth regions, and that this adaptive mixture generalizes across different PDE types.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"U-HNO uses adaptive per-point routing in a U-shaped hybrid architecture to achieve state-of-the-art accuracy on PDE benchmarks with sharp localized features.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"U-HNO uses per-point hard masks to route between global Fourier and local Gaussian branches for PDEs with mixed smooth and sharp dynamics.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"69537e2f141d8b2a0d0a9274d90c02b98e2ac2fbeedc87548953d34cb6d74413"},"source":{"id":"2605.12965","kind":"arxiv","version":1},"verdict":{"id":"6fb716f1-ec4d-4165-8ab9-0bcb80dea57e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T20:27:29.619094Z","strongest_claim":"Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features.","one_line_summary":"U-HNO uses adaptive per-point routing in a U-shaped hybrid architecture to achieve state-of-the-art accuracy on PDE benchmarks with sharp localized features.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that a per-pixel hard mask based on local contrast of the routing signal can reliably select the appropriate branch (global or local) without causing training instabilities or degrading performance on smooth regions, and that this adaptive mixture generalizes across different PDE types.","pith_extraction_headline":"U-HNO uses per-point hard masks to route between global Fourier and local Gaussian branches for PDEs with mixed smooth and sharp dynamics."},"references":{"count":30,"sample":[{"doi":"","year":2013,"title":"Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation","work_id":"1fe8c7c8-aff7-4b94-9096-e549d7e60789","ref_index":1,"cited_arxiv_id":"1308.3432","is_internal_anchor":true},{"doi":"","year":2023,"title":"Spherical fourier neural operators: Learning stable 9 dynamics on the sphere","work_id":"2af53f4b-30d6-416f-91df-9efc22c39900","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"Message passing neural pde solvers","work_id":"6a58fccd-2012-4c4a-88ae-8ccf1a2a4d3c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Choose a transformer: Fourier or galerkin.Advances in neural information processing systems, 34:24924–24940, 2021","work_id":"a93e6c14-a11d-43be-a910-3c6c38b91c87","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Gupta, G., Xiao, X., and Bogdan, P","work_id":"13d52677-f17c-4a4a-8e65-9b2670b49e58","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"9bbf90081c661b26d4214b5169d77df0d93bb0191d51c65b2b736430be81672d","internal_anchors":5},"formal_canon":{"evidence_count":2,"snapshot_sha256":"574f43658a9ae17096f4d04ffffadd1e376764c95588dd57492c129414395428"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}