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.

翻译：参数化偏微分方程解流形的Kolmogorov n-宽度缓慢衰减，导致难以实现基于线性投影的高效降阶模型。这是因为需要高维度的缩减空间才能以足够精度逼近解流形。为解决这一问题，不同架构的神经网络被用于构建解流形的精确非线性回归。然而，大多数实现属于非侵入式黑箱代理模型，仅部分模型能将离散化参数模型的自由度数量降至潜在维度。本文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/在线阶段不抛弃底层的物理与数值模型。我们重点研究自编码器，用于进一步压缩解流形线性逼近器的维度，最终实现非线性降维。在获得精确的非线性逼近器后，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，并通过恰当的超缩减策略使其与自由度数量解耦。结合局部非线性逼近器的使用，我们发展了新的自适应超缩减策略。该方法在两个非线性含时参数基准问题上进行验证：涉及可变马赫数下NACA翼型周围的超声速流动，以及可变倾斜角下Ahmed体周围的不可压缩湍流。