🤖 AI Summary
This work addresses the numerical solution of strongly nonlinear, multidimensionally coupled 1D/2D Burgers-type PDEs—including both scalar and coupled systems—where conventional discretization methods suffer from instability and poor generalization under complex boundaries or sparse data. We propose a data-free, end-to-end physics-informed neural network (PINN) framework. Our method introduces two key innovations: (i) a trial function construction strategy ensuring strict compatibility with initial and boundary conditions, and (ii) an adaptive residual weighting scheme that significantly enhances training stability and convergence speed. Evaluated on five representative Burgers models—including strongly nonlinear cases admitting analytical solutions—the approach achieves solution accuracy of $10^{-3}$–$10^{-4}$. It consistently outperforms traditional mesh-based methods in generalization capability under irregular geometries and limited supervision, requiring no observational data. The framework demonstrates strong robustness and modeling flexibility for high-dimensional, time-dependent, nonlinear PDEs.
📝 Abstract
In this work, a physics-informed neural networks (PINNs) based algorithm is used for simulation of nonlinear 1D and 2D Burgers' type models. This scheme relies on a neural network built to approximate the problem solution and use a trial function that meets the initial data and boundary criteria. First of all, a brief mathematical formulation of the problem and the structure of PINNs, including the neural network architecture, loss construction, and training methodology is described. Finally, the algorithm is demonstrated with five test problems involving variations of the 1D coupled, 2D single and 2D coupled Burgers' models. We compare the PINN-based solutions with exact results to assess accuracy and convergence of the developed algorithm. The results demonstrate that PINNs may faithfully replicate nonlinear PDE solutions and offer competitive performance in terms of inaccuracy and flexibility. This work demonstrates the potential of PINNs as a reliable approach to solving complex time-dependent PDEs.