Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems
2026-03-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address difficulties in simulating how things move and change quickly in fluids or materials, especially when sharp changes happen. They combine a traditional mathematical method (SUPG with shock-capturing) and a neural network (PINN) to improve accuracy near the end of the simulation instead of throughout the whole process. Their hybrid method selectively focuses the neural network where it's most needed, leading to better results in several test cases compared to using the traditional method alone. This approach helps capture sharp features like waves and layers more accurately in time-dependent problems.
Convection-diffusion-reaction equationsStreamline-Upwind Petrov-Galerkin (SUPG)Shock-capturingPhysics-informed neural networks (PINNs)Transient transport phenomenaFinite element methodsResidual blocksBurgers equationAdaptive loss weightingNumerical stabilization
Authors
Süleyman Cengizci, Ömür Uğur, Srinivasan Natesan
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.