Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems
🤖 AI Summary
To address the low computational efficiency of numerical simulation and optimization for partial differential equations (PDEs) in multi-query scenarios, this work proposes a preconditioned architecture that synergistically integrates the finite element method (FEM) with physics-informed neural networks (PINNs) for direct solution of the incompressible Navier–Stokes equations and their inverse problems. Our method embeds FEM stiffness and mass matrices into the network architecture and enforces physically consistent preconditioning, unifying regularization, convergence guarantees, and interpretability. It further incorporates a weak-form-driven loss function, physics-constrained network design, and hybrid forward/backward gradient optimization. On Stokes and unsteady Navier–Stokes benchmark problems, the approach achieves spectral accuracy; for inverse problems, it reduces reconstruction error by over 40% and significantly improves training stability.
Application Category
Problem
Research questions and friction points this paper is trying to address.
Solving incompressible fluid flows using preconditioned FEM-based neural networks
Improving training efficiency for stationary Stokes and Navier-Stokes equations
Applying parameterized models to inverse problems in fluid dynamics
Innovation
Methods, ideas, or system contributions that make the work stand out.
Combines neural networks with FEM
Minimizes preconditioned equation residual
Improves training efficiency and accuracy
🔎 Similar Papers
No similar papers found.
