An efficient hp-Variational PINNs framework for incompressible Navier-Stokes equations
这是一篇使用 VPinn 来求解 Navier-Stokes 方程的论文。
我们先来看看这篇论文的摘要：
摘要
英文原文：
Physics-informed neural networks (PINNs) are able to solve partial differential equations (PDEs) by incorporating the residuals of the PDEs into their loss functions. Variational Physics-Informed Neural Networks (VPINNs) and hpVPINNs use the variational form of the PDE residuals in their loss function. Although hp-VPINNs have shown promise over traditional PINNs, they suffer from higher training times and lack a framework capable of handling complex geometries, which limits their application to more complex PDEs. As such, hp-VPINNs have not been applied in solving the Navier-Stokes equations, amongst other problems in CFD, thus far. FastVPINNs was introduced to address these challenges by incorporating tensor-based loss computations, significantly improving the training efficiency. Moreover, by using the bilinear transformation, the FastVPINNs framework was able to solve PDEs on complex geometries. In the present work, we extend the FastVPINNs framework to vector-valued problems, with a particular focus on solving the incompressible Navier-Stokes equations for two-dimensional forward and inverse problems, including problems such as the lid-driven cavity flow, the Kovasznay flow, and flow past a backward-facing step for Reynolds numbers up to 200. Our results demonstrate a 2x improvement in training time while maintaining the same order of accuracy compared to PINNs algorithms documented in the literature. We further showcase the framework’s efficiency in solving inverse problems for the incompressible Navier-Stokes equations by accurately identifying the Reynolds number of the underlying flow. Additionally, the framework’s ability to handle complex geometries highlights its potential for broader applications in computational fluid dynamics. This implementation opens new avenues for research on hp-VPINNs, potentially extending their applicability to more complex problems.
翻译：
物理信息的神经网络（PINN）能够通过将 PDE 的残差纳入其损失函数来解决部分微分方程（PDE）。变异物理信息的神经网络（VPINN）和 HPVPINNS 在其损失函数中使用 PDE 残差的变异形式。尽管 HP-vpinns 对传统的 PINN 表现出了希望，但它们遭受了较高的训练时间，并且缺乏能够处理复杂几何形状的框架，从而将其应用限制在更复杂的 PDES 中。因此，迄今为止，尚未应用 HP-VPINN 在求解 Navier-Stokes 方程中，以及 CFD 中的其他问题。引入了 FASTVPINNS，通过结合基于张量的损失计算，从而显着提高训练效率，以应对这些挑战。此外，通过使用双线性转换，FastVpinns 框架能够在复杂的几何形状上求解 PDE。在目前的工作中，我们将 FASTVPINNS 框架扩展到了矢量值问题，特别着眼于解决不可压缩的 Navier-Stokes 方程，以解决二维向前和反向问题，包括诸如盖子驱动的腔流，kovasznay 流量以及以后的阶段进行阶段的阶段，以改进阶段的时间，并将其进行了训练。与文献中记录的 PINNS 算法相比，准确性。我们通过准确识别基础流的雷诺数数量，进一步展示了该框架在解决不可压缩的 Navier-Stokes 方程中的反问题方程的效率。此外，该框架处理复杂几何形状的能力突出了其在计算流体动力学中更广泛应用的潜力。该实施为 HP-VPINNS 研究开辟了新的途径，有可能将其适用性扩展到更复杂的问题上。