Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems

TL;DR

This work presents a hybrid computational framework that extends the PINN-Augmented SUPG with Shock-Capturing (PASSC) methodology from steady to unsteady problems and combines a semi-discrete stabilized finite element method with a PINN-based correction strategy for transient convection-diffusion-reaction equations.

Abstract

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.

Figures (19)

• Figure 1: Flowchart illustrating the selective physics enforcement strategy. At each training iteration, spatiotemporal collocation points $(t_j, \mathbf{x}_j)$ are sampled uniformly at random within the training time window and spatial domain. Only points whose spatial distance to the boundary satisfies $d_{\partial\Omega}(\mathbf{x}_j) > d_\text{min}$ and for which the minimum count criterion $N_\text{int} \geq N_\text{int}^\text{min}$ holds are retained for PDE residual computation. Boundary-adjacent points are excluded to preserve numerical stability near sharp gradients. The strong-form residual, including the temporal derivative $\partial u_\theta / \partial t$ obtained via automatic differentiation, is clamped to $[-C, C]$ before computing the loss to prevent gradient explosion.
• Figure 2: Schematic of the PASSC neural network architecture (shown for $n_h=128$, $n_r=8$, $n_\text{F}=24$, $n_\text{sd}=2$). The spatiotemporal input $\mathbf{z}=(t,x_1,x_2)\in\mathbb{R}^{3}$ is mapped through a random Fourier feature embedding to $\boldsymbol{\varphi}(\mathbf{z})\in\mathbb{R}^{51}$, projected to a hidden dimension of $128$ via the input layer with SiLU activation, and then processed by $8$ residual blocks---each comprising two fully connected layers, a skip (residual) connection, and layer normalization. A narrowing layer ($128\!\to\!64$) followed by a single output neuron produces the predicted solution $u_\mathrm{NN}(t,\mathbf{x})$.
• Figure 3: Comparison of the numerical solutions obtained by SUPG, SUPG-YZ$\beta$, and the hybrid PINN with the analytical solution for Example 1 with $\varepsilon = 10^{-4}$: (\ref{['fig:fig1a']}) entire domain and (\ref{['fig:fig1b']}) zoom in the boundary layer region near $x = 1$.
• Figure 4: Error analysis for Example 1 with $\varepsilon = 10^{-4}$: (\ref{['fig:fig2a']}) evolution of the $L^2$ error over the time interval $[0, 1]$ for SUPG and SUPG-YZ$\beta$, with the hybrid PINN error at the terminal time $t_\text{f} = 1$ indicated by the star marker; (\ref{['fig:fig2b']}) pointwise absolute error at $t_\text{f} = 1$ across the spatial domain.
• Figure 5: Training diagnostics for Example 1 with $\varepsilon = 10^{-4}$: (\ref{['fig:fig3a']}) evolution of the individual loss components (total, data, PDE, and BC) over $5000$ training epochs on a logarithmic scale; (\ref{['fig:fig3b']}) adaptive weight schedule $w_{\text{data}}$, $w_{\text{pde}}$, and $w_{\text{bc}}$ governing the three-phase training strategy.

Theorems & Definitions (20)

• Remark 1
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