可解释的超降阶模型在可压缩与不可压缩Navier-Stokes方程非线性近似解流形上的应用
Explicable hyper-reduced order models on nonlinearly approximated solution manifolds of compressible and incompressible Navier-Stokes equations
AI总结：
本文提出一种基于自编码器非线性降维与残差最小二乘Petrov-Galerkin超降阶的侵入式可解释降阶模型方法，并在NACA翼型超声速流和Ahmed体湍流基准上验证其有效性。
摘要：
参数偏微分方程解流形的Kolmogorov n-宽度缓慢衰减，阻碍了基于线性投影的高效降阶模型的实现。这是因为为了以足够精度近似解流形，所需的降阶空间维数过高。为解决这一问题，不同架构的神经网络已被用于构建解流形的精确非线性回归。然而，大多数实现是非侵入式的黑箱代理模型，只有一部分能够将离散参数模型的自由度数降维到潜在维度。我们提出一种新的侵入式且可解释的降阶建模方法，该方法采用神经网络进行解流形近似，但在预测/在线阶段不丢弃底层的物理模型和数值模型。我们将重点放在自编码器上，用于进一步压缩解流形线性近似的维数，最终实现非线性降维。在获得精确的非线性近似之后，我们采用基于残差的非线性最小二乘Petrov-Galerkin方法在潜在流形上求解，并适当进行超降阶处理，使其独立于自由度数。我们开发了新的自适应超降阶策略，并结合局部非线性近似进行应用。我们在两个非线性时间相关参数基准上测试了该方法，包括马赫数变化的超声速流绕过NACA翼型，以及斜角变化的Ahmed体周围不可压缩湍流。
英文摘要：
A slow decaying Kolmogorov n-width of the solution manifold of a parametric partial differential equation precludes the realization of efficient linear projection-based reduced-order models. This is due to the high dimensionality of the reduced space needed to approximate with sufficient accuracy the solution manifold. To solve this problem, neural networks, in the form of different architectures, have been employed to build accurate nonlinear regressions of the solution manifolds. However, the majority of the implementations are non-intrusive black-box surrogate models, and only a part of them perform dimension reduction from the number of degrees of freedom of the discretized parametric models to a latent dimension. We present a new intrusive and explicable methodology for reduced-order modelling that employs neural networks for solution manifold approximation but that does not discard the physical and numerical models underneath in the predictive/online stage. We will focus on autoencoders used to compress further the dimensionality of linear approximants of solution manifolds, achieving in the end a nonlinear dimension reduction. After having obtained an accurate nonlinear approximant, we seek for the solutions on the latent manifold with the residual-based nonlinear least-squares Petrov-Galerkin method, opportunely hyper-reduced in order to be independent from the number of degrees of freedom. New adaptive hyper-reduction strategies are developed along with the employment of local nonlinear approximants. We test our methodology on two nonlinear time-dependent parametric benchmarks involving a supersonic flow past a NACA airfoil with changing Mach number and an incompressible turbulent flow around the Ahmed body with changing slant angle.
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