FFV-PINN: A Fast Physics-Informed Neural Network with Simplified Finite Volume Discretization and Residual Correction

2026-03-25

AI Summary

This work addresses the limitations of conventional physics-informed neural networks (PINNs)—notably insufficient physical constraints, training instability, and slow convergence—stemming from their neglect of neighborhood information, which hinders their ability to solve complex fluid problems at high Reynolds or Rayleigh numbers. To overcome these challenges, we propose the Fast Finite Volume Physics-Informed Neural Network (FFV-PINN), which uniquely integrates a simplified finite volume method and a residual correction mechanism into the PINN framework. By employing a simplified discretization of convective terms and directly approximating cell-face values through the network output, FFV-PINN enhances both physical consistency and computational efficiency. The method markedly improves dispersion and dissipation properties, enabling accurate and rapid data-free solutions of challenging flows: two- and three-dimensional lid-driven cavity flows at Re = 10⁴ (solved in 680 seconds) and natural convection at Ra = 10⁸ (231 seconds)—the first successful PINN-based solutions for such high-difficulty cases.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Learning with Manifolds

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

Physics-informed neural networks (PINNs) have emerged as a major research focus. However, today's PINNs encounter several limitations. Firstly, during the construction of the loss function using automatic differentiation, PINNs often neglect information from neighboring points, which hinders their ability to enforce physical constraints and diminishes their accuracy. Furthermore, issues such as instability and poor convergence persist during PINN training, limiting their applicability to complex fluid dynamics problems. To address these challenges, a fast physics-informed neural network framework that integrates a simplified finite volume method (FVM) and residual correction loss term has been proposed, referred to as Fast Finite Volume PINN (FFV-PINN). FFV-PINN utilizes a simplified FVM discretization for the convection term, with an accompanying improvement in the dispersion and dissipation behavior. Unlike traditional FVM, the FFV-PINN outputs can be simply and directly harnessed to approximate values on control surfaces, thereby simplifying the discretization process. Moreover, a residual correction loss term is introduced to significantly accelerates convergence and improves training efficiency. To validate the performance, we solve a series of challenging problems -- including flow in the two-dimensional steady and unsteady lid-driven cavity, three-dimensional steady lid-driven cavity, backward-facing step flows, and natural convection at high Reynolds number and Rayleigh number. Notably, the FFV-PINN can achieve data-free solutions for the lid-driven cavity flow at Re = 10000 and natural convection at Ra = 1e8 for the first time in PINN literature, even while requiring only 680s and 231s. It further highlights the effectiveness of FFV-PINN in improving both speed and accuracy, marking another step forward in the progression of PINNs as competitive neural PDE solvers.

Problem

• Physics-informed neural networks
• Finite volume method
• Convergence
• Stability
• Fluid dynamics

Innovation

• Physics-Informed Neural Networks
• Finite Volume Method
• Residual Correction
• Convection Term Discretization
• High Reynolds Number Flow