Gradient Driven Physics Informed Neural Networks for Conduction Heat Transfer and Incompressible Laminar Flow

Abstract Physics-Informed Neural Networks (PINNs) have opened new possibilities for solving partial differential equations (PDEs) by embedding physical laws directly into the learning process. However, despite their flexibility, traditional PINNs often struggle to capture sharp gradients and intricate solution features, which limits their effectiveness in many practical problems. In this work, we have introduced Gradient-Driven Physics-Informed Neural Networks (GDPINNs) that improve the ability of traditional PINNs to resolve sharp gradients. By incorporating gradient information directly into the loss function, GDPINNs better target regions where traditional PINNs typically fail. We validated the method on steady-state and transient heat conduction problems, including a central heating source and a sinusoidal boundary condition, and found strong agreement with reference solutions. To further understand the framework's capability, we applied it to a high-gradient steady-state and transient heat conduction problem, where GDPINNs show clear advantages over traditional PINNs and align closely with reference results. We also extended GDPINNs to incompressible laminar flow in a lid-driven cavity, demonstrating its broader applicability. In these cases, GDPINNs consistently provide higher accuracy and better capture critical solution features, highlighting their potential to improve PINNs-based approaches for complex physical problems with sharp gradients.