AI 中文总结
针对固定参数松弛PINNs无法跟踪多尺度过渡的问题，提出JXRGCM方法，将松弛参数退火至零，使标量守恒律的L²收敛速率达O(ε^(1/4))，并在数值实验中提升了激波与稀疏波的分辨能力。
AI 中文摘要
非线性双曲守恒律的Jin-Xin松弛引入了一个松弛参数，该参数控制着激波分辨的内部层宽度；当该宽度消失时，极限守恒律的不连续性仅在奇异极限中出现。物理信息神经网络（PINNs）使用平滑的网络近似，因此并不适合该极限，而具有固定参数的松弛PINNs仅能分辨单一尺度，无法跟踪向极限解的多尺度过渡。我们提出了Jin-Xin松弛渐进收敛方法（JXRGCM），该方法将松弛参数视为延拓变量，按计划将其退火至零，并从先前阶段热启动每个阶段，使得近似解通过逐渐更尖锐的尺度跟踪松弛剖面。在亚特征条件下，我们建立了一个稳定性估计，其常数与松弛参数无关；结合松弛极限，对于标量守恒律，该方法向熵解的L²收敛速率达到O(ε^(1/4))。针对Burgers方程、浅水波溃坝问题和Sod激波管的数值实验表明，与固定参数松弛PINNs及其他物理信息方法相比，JXRGCM提升了激波和稀疏波的分辨能力。
英文摘要
The Jin--Xin relaxation of a nonlinear hyperbolic conservation law introduces a relaxation parameter that controls the width of the internal layer resolving a shock; the discontinuity of the limiting conservation law emerges only in the singular limit as this width vanishes. Physics-informed neural networks (PINNs) use smooth network approximations and are therefore not well suited to this limit, while relaxation PINNs with a fixed parameter resolve only a single scale and cannot follow the multiscale transition toward the limiting solution. We propose the Jin--Xin relaxation gradual convergence method (JXRGCM), which treats the relaxation parameter as a continuation variable, annealing it to zero along a schedule and warm-starting each stage from the previous one, so that the approximation follows the relaxation profile through progressively sharper scales. Under the sub-characteristic condition we establish a stability estimate whose constant is independent of the relaxation parameter; combined with the relaxation limit, it yields for scalar conservation laws an $L^2$ convergence rate of $\mathcal{O}(\varepsilon^{1/4})$ toward the entropy solution. Numerical experiments on the Burgers equation, the shallow-water dam-break problem, and the Sod shock tube show that JXRGCM improves shock and rarefaction resolution compared with fixed-parameter relaxation PINNs and other physics-informed approaches.
