用物理信息神经网络模拟非线性波方程，精度高且灵活。
PINNs Algorithmic Framework for Simulation of Nonlinear Burgers' Type Models
- 构建神经网络逼近解，通过试函数满足初值与边界条件。
- 五组测试显示结果接近精确解，误差小且收敛稳定。
- 适合求解复杂时变偏微分方程，尤其适用于缺乏数据场景。
本文提出一种基于物理信息神经网络（PINNs）的算法，用于求解一维和二维非线性Burgers型模型。该方法通过神经网络近似问题解，并设计满足初始条件与边界条件的试函数。首先简要介绍问题的数学表述及PINNs结构，包括神经网络架构、损失函数构建与训练方法。随后，通过五个测试案例验证算法性能，涵盖一维耦合、二维单个及二维耦合Burgers模型。将PINN求解结果与精确解对比，评估其精度与收敛性。结果表明，PINNs能准确复现非线性偏微分方程解，在误差和灵活性方面表现优异，展现出作为求解复杂时变偏微分方程可靠方法的巨大潜力。
原文摘要 · Abstract (English)
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.
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