🤖 AI Summary
This work addresses the challenge that standard physics-informed neural networks (PINNs) struggle to accurately approximate solutions and their derivatives in convection-dominated problems featuring thin-layer structures. To overcome this, the authors propose LRX-PINN, a method that constructs basis functions with transition-like behavior by incorporating Cauchy activation functions. These basis functions exhibit transitional profiles in the solution space, while their derivatives correspond to localized Cauchy kernels that precisely align with the scale of thin layers. The resulting neural representation is structurally consistent with the underlying solution, offering both derivative stability and exponential approximation capability. The study further reveals that the effective physical width of ridge neurons is $d/|w|$. Implemented within an XNet architecture and embedded into an hp-VPINN framework, LRX-PINN achieves higher accuracy than existing methods using fewer than 30% of their parameters, and further surpasses state-of-the-art performance when integrated into the hp-VPINN framework.
📝 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.