论文标题
有效的非线性歧管降低订单模型
Efficient nonlinear manifold reduced order model
论文作者
论文摘要
传统的线性子空间还原订单模型（LS-ROM）能够加速物理模拟，其中固有的解决方案空间落入具有较小尺寸的子空间，即，解决方案空间的kolmogorov n脚步很小。但是，对于不适合这种类型的物理现象，例如，平流为主的流动现象，低维线性子空间近似于溶液。为了解决此类案例，我们开发了有效的非线性歧管ROM（NM-ROM），该案例可以更好地近似于比LS-ROM较小的潜在模型解决方案。我们的方法利用了用于求解相应的全订单模型（FOM）的现有数值方法。通过在NM-ROM的背景下开发超还原技术来实现效率。数值结果表明，神经网络可以从较高的雷诺数的2D汉堡方程中学习更有效的潜在空间表示。通过过度还原技术对非线性术语进行适当的处理，可以实现2D汉堡方程的速度高达11.7。
Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, such as advection-dominated flow phenomena, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed an efficient nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LS-ROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models (FOMs). The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 2D Burgers' equations with a high Reynolds number. A speed-up of up to 11.7 for 2D Burgers' equations is achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique.
