Hypernetwork-Conditioned WENO5 Conservative-Form CNNs for One-Dimensional Conservation Laws
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| Format: | Preprint |
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2026
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| author | Chen, Yongsheng Guo, Wei Zhong, Xinghui |
| author_facet | Chen, Yongsheng Guo, Wei Zhong, Xinghui |
| contents | We study a conservative data-driven discretization for one-dimensional hyperbolic conservation laws based on the classical fifth-order WENO finite-volume scheme and a hypernetwork architecture. In the proposed Hyper--WENO5 Conservative-Form Convolutional Neural Network (Hyper--CFCNN), a lightweight target network predicts the nonlinear WENO weights on each stencil, while a hypernetwork generates the target-network parameters from problem metadata, including the mesh spacing, mesh layout, and coarse descriptors of the initial condition. The construction preserves the standard polynomial reconstruction and conservative flux-difference update of WENO, which enables adaptation across problem instances and spatial resolutions without retraining. We also consider an unknown-flux variant, Hyper--CFCNN--F, in which a compact FluxNet is used in place of the analytical flux inside the numerical flux function while retaining a conservative update form. To improve long-time prediction quality, training uses a multi-step recurrent loss that penalizes error accumulation over successive time advances. Numerical experiments on one-dimensional test problems, including single- and multi-shock Burgers equations, the shallow-water system, and the Shu--Osher Euler example, show that Hyper--CFCNN attains accuracy comparable to classical WENO5, achieves near machine-precision conservation in the known-flux setting on fine meshes, and generalizes to unseen spatial resolutions and initial conditions without retraining. The flux-learning variant remains stable on meshes outside the training set and exhibits bounded conservation drift. These results show that hypernetwork-conditioned conservative WENO discretizations provide an effective framework for adaptive high-order learning of nonlinear conservation laws with either known or unknown fluxes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13106 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypernetwork-Conditioned WENO5 Conservative-Form CNNs for One-Dimensional Conservation Laws Chen, Yongsheng Guo, Wei Zhong, Xinghui Numerical Analysis 35L65, 65M08, 68T07 We study a conservative data-driven discretization for one-dimensional hyperbolic conservation laws based on the classical fifth-order WENO finite-volume scheme and a hypernetwork architecture. In the proposed Hyper--WENO5 Conservative-Form Convolutional Neural Network (Hyper--CFCNN), a lightweight target network predicts the nonlinear WENO weights on each stencil, while a hypernetwork generates the target-network parameters from problem metadata, including the mesh spacing, mesh layout, and coarse descriptors of the initial condition. The construction preserves the standard polynomial reconstruction and conservative flux-difference update of WENO, which enables adaptation across problem instances and spatial resolutions without retraining. We also consider an unknown-flux variant, Hyper--CFCNN--F, in which a compact FluxNet is used in place of the analytical flux inside the numerical flux function while retaining a conservative update form. To improve long-time prediction quality, training uses a multi-step recurrent loss that penalizes error accumulation over successive time advances. Numerical experiments on one-dimensional test problems, including single- and multi-shock Burgers equations, the shallow-water system, and the Shu--Osher Euler example, show that Hyper--CFCNN attains accuracy comparable to classical WENO5, achieves near machine-precision conservation in the known-flux setting on fine meshes, and generalizes to unseen spatial resolutions and initial conditions without retraining. The flux-learning variant remains stable on meshes outside the training set and exhibits bounded conservation drift. These results show that hypernetwork-conditioned conservative WENO discretizations provide an effective framework for adaptive high-order learning of nonlinear conservation laws with either known or unknown fluxes. |
| title | Hypernetwork-Conditioned WENO5 Conservative-Form CNNs for One-Dimensional Conservation Laws |
| topic | Numerical Analysis 35L65, 65M08, 68T07 |
| url | https://arxiv.org/abs/2605.13106 |