{"ID":5937878,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-07T03:14:33.014478982Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03682","arxiv_id":"2607.03682","title":"LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems","abstract":"Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--derivative structure of such layers. We propose a Layer-Resolving XNet Physics-Informed Neural Network (LRX-PINN) based on integrated Cauchy activations. The proposed basis is transition-type at the solution level, while its derivative recovers a localized Cauchy kernel. We show that this structure matches the scaling of convection-dominated layers, inherits the Cauchy approximation mechanism at the derivative-profile level, and identifies \\(d/\\|w\\|\\) as the effective physical width of a ridge neuron. For analytic layer profiles, this yields derivative-stable exponential approximation in the stretched coordinate and a layer-scaled estimate for the strong residual of the singularly perturbed operator. Numerical experiments on several convection-dominated benchmarks show that LRX-PINN achieves higher accuracy than PIKAN and Fourier-feature PINNs while using less than \\(30\\%\\) of their trainable parameters. On more challenging benchmarks, embedding the proposed representation into hp-VPINN-based frameworks further improves the best results obtained by existing hp-VPINN-based baselines without changing their original loss functionals or stabilization strategies. These results show that neural representations aligned with layer structure provide a compact and effective approach for convection-dominated problems.","short_abstract":"Convection-dominated convection-diffusion problems often develop thin layers, where the solution has sharp transition profiles and its derivatives are highly localized. This creates a structural mismatch for standard physics-informed neural networks (PINNs), whose trial spaces are not designed to match the value--deriv...","url_abs":"https://arxiv.org/abs/2607.03682","url_pdf":"https://arxiv.org/pdf/2607.03682v1","authors":"[\"Zihao Guo\",\"Xin Li\",\"Zhihong Xia\"]","published":"2026-07-04T03:25:05Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"cs.LG\"]","methods":"[\"Diffusion Model\"]","has_code":false}