发表机构
Institute for Advanced Research, Great Bay University; College of Computer Science and Technology, Dongguan University of Technology; Department of Mathematics, Northwestern University（先进研究院，大湾大学; 计算机科学与技术学院，东莞理工大学; 数学系，西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对对流主导的对流扩散问题中标准物理信息神经网络的结构不匹配，提出基于集成柯西激活的LRX-PINN，其结构匹配对流主导层缩放，实验表明该方法参数少精度高。
AI 中文摘要
对流主导的对流扩散问题常出现薄层，标准物理信息神经网络结构不匹配。我们提出基于集成柯西激活的层解析XNet物理信息神经网络（LRX-PINN）。其结构匹配对流主导层缩放，继承柯西近似机制，识别有效物理宽度。数值实验表明其精度高、参数少，嵌入hp-VPINN框架可进一步提升效果。
英文摘要
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.
Comments30 pages; 16 figures
