Design principles for stable and generalizable data-driven discretizations for solving linear hyperbolic conservation laws

Mathematics > Numerical Analysis [Submitted on 16 Jun 2026] Title:Design principles for stable and generalizable data-driven discretizations for solving linear hyperbolic conservation laws View PDF HTML (experimental)Abstract:We investigate data-driven finite-volume discretizations of the linear advection equation in one dimension. Neural networks for use as numerical advection schemes are constructed adhering to first principles of numerical analysis, allowing us to examine how normalization, training data, and architectural choices influence stability, accuracy, and shape preservation. (i) We show that reconstruction based solely on cell averages leads to a multi-valued learning problem, explaining limited generalization when training data includes widely different curvature regimes. (ii) Numerical stability and good generalization can be achieved by enforcing semilinearity (Lin and Rood 1998) through local stencil-scale normalization, which ensures invariance under affine transformations of the inputs. (iii) A new data-driven flux limiter is introduced that outperforms the classical 'OSTVD3' (Arora and Roe, 1997) scheme in shape preservation by introducing mild antidiffusion in near-linear regimes, while higher-order reconstruction in non-monotonic regions provides limited benefit. (iv) We show that training on polynomial profiles yields stable, high-order accurate discretizations, with the polynomial degree controlling the formal order of accuracy. Together, these results illustrate how the representational, architectural, and training choices govern the stability and generalization of data-driven finite-volume schemes for linear advection. Submission history From: Antoine Alexis Nasser [view email][v1] Tue, 16 Jun 2026 04:13:03 UTC (6,494 KB) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.