The numerical simulation of convection-dominated transient transport phenomena poses significant computational challenges due to sharp gradients and propagating fronts across the spatiotemporal domain. Classical discretization methods often generate spurious oscillations, requiring advanced stabilization techniques. However, even stabilized finite element methods may require additional regularization to accurately resolve localized steep layers. On the other hand, standalone physics-informed neural networks (PINNs) struggle to capture sharp solution structures in convection-dominated regimes and typically require a large number of training epochs. This work presents a hybrid computational framework that extends the PINN-Augmented SUPG with Shock-Capturing (PASSC) methodology from steady to unsteady problems. The approach combines a semi-discrete stabilized finite element method with a PINN-based correction strategy for transient convection-diffusion-reaction equations. Stabilization is achieved using the Streamline-Upwind Petrov-Galerkin (SUPG) formulation augmented with a YZbeta shock-capturing operator. Rather than training over the entire space-time domain, the neural network is applied selectively near the terminal time, enhancing the finite element solution using the last K_s temporal snapshots while enforcing residual constraints from the governing equations and boundary conditions. The network incorporates residual blocks with random Fourier features and employs progressive training with adaptive loss weighting. Numerical experiments on five benchmark problems, including boundary and interior layers, traveling waves, and nonlinear Burgers dynamics, demonstrate significant accuracy improvements at the terminal time compared to standalone stabilized finite element solutions.

翻译：瞬态对流主导输运现象的数值模拟在时空域中面临急剧梯度和传播前沿带来的显著计算挑战。经典离散化方法常产生伪振荡，需采用先进稳定化技术。然而，即使稳定化有限元方法也可能需要额外正则化以精确解析局部陡峭层。另一方面，独立运行的物理信息神经网络（PINN）难以捕捉对流主导区域中的急剧解结构，且通常需要大量训练轮次。本研究提出一种混合计算框架，将带激波捕捉的PINN增强SUPG（PASSC）方法从稳态问题扩展至非稳态问题。该方法将半离散稳定化有限元方法与基于PINN的瞬态对流-扩散-反应方程修正策略相结合。稳定化通过采用经YZbeta激波捕捉算子增强的流线迎风Petrov-Galerkin（SUPG）格式实现。神经网络并非在整个时空域训练，而是选择性地应用于终端时间附近，利用最后K_s个时间快照增强有限元解，同时强制满足控制方程和边界条件的残差约束。该网络集成具有随机傅里叶特征的残差块，并采用自适应损失加权的渐进式训练。在包含边界层与内层、行波和非线性Burgers动力学在内的五个基准问题上进行的数值实验表明，相较于独立稳定化有限元解，该方法在终端时间能实现显著的精度提升。