PhysicsFormer: An Efficient and Fast Attention-Based Physics Informed Neural Network for Solving Incompressible Navier Stokes Equations

Abstract: Traditional experimental and numerical approaches for fluid dynamics problems often suffer from high computational cost, mesh sensitivity, and limited capability in capturing complex physical behaviors. Moreover, conventional physics-informed neural networks (PINNs) frequently struggle in chaotic and highly unsteady flow regimes. In this work, we propose \textit{PhysicsFormer}, a fast and efficient transformer-based physics-informed framework that incorporates multi-head encoder-decoder cross-attention. Unlike multilayer perceptron-based PINNs, PhysicsFormer operates on sequential representations constructed from spatio-temporal data, enabling effective learning of long-range temporal dependencies and improved propagation of initial condition information. A data-embedding strategy is employed to convert spatio-temporal points into pseudo-sequences, while a dynamics-weighted loss function replaces the standard PINNs formulation. Owing to its parallel learning structure, PhysicsFormer demonstrates superior computational efficiency compared to existing transformer-based approaches. The framework is validated on Burgers' equation and flow reconstruction governed by the Navier-Stokes equations, achieving mean squared errors on the order of $10^{-6}$. In addition, an inverse problem involving parameter identification in the two-dimensional incompressible Navier-Stokes equations is investigated. For clean data, PhysicsFormer achieves zero identification error for both $λ_1$ and $λ_2$; under $1\%$ Gaussian noise, the errors are $0.07\%$ for $λ_1$ and $0\%$ for $λ_2$. These results demonstrate that PhysicsFormer provides a reliable and computationally efficient surrogate modeling framework for time-dependent fluid flow problems.