Many engineering applications rely on the evaluation of expensive, non-linear
high-dimensional functions. In this paper, we propose the RONAALP algorithm
(Reduced Order Nonlinear Approximation with Active Learning Procedure) to
incrementally learn a fast and accurate reduced-order surrogate model of a
target function on-the-fly as the application progresses. First, the
combination of nonlinear auto-encoder, community clustering and radial basis
function networks allows to learn an efficient and compact surrogate model with
limited training data. Secondly, the active learning procedure overcome any
extrapolation issue when evaluating the surrogate model outside of its initial
training range during the online stage. This results in generalizable, fast and
accurate reduced-order models of high-dimensional functions. The method is
demonstrated on three direct numerical simulations of hypersonic flows in
chemical nonequilibrium. Accurate simulations of these flows rely on detailed
thermochemical gas models that dramatically increase the cost of such
calculations. Using RONAALP to learn a reduced-order thermodynamic model
surrogate on-the-fly, the cost of such simulation was reduced by up to 75%
while maintaining an error of less than 10% on relevant quantities of interest.

提出 RONAALP 算法用于增量学习快速准确的高维函数的降阶代理模型，在线阶段使用主编码器、社区聚类和径向基函数网络相结合的方式学习高效紧凑的代理模型，通过主动学习程序克服模型在初始训练范围外进行外插的问题，从而得到能够推广且快速准确的高维函数的降阶模型，并在化学非平衡的高超声速流动的三个数值模拟中加以验证，通过动态学习降阶热力学模型代理，将模拟计算的成本减少了 75%，同时保持感兴趣物理量上的误差小于 10%。

RONAALP：带有主动学习过程的降阶非线性逼近

RONAALP: Reduced-Order Nonlinear Approximation with Active Learning  Procedure

High-fidelity numerical simulations of partial differential equations (PDEs)
given a restricted computational budget can significantly limit the number of
parameter configurations considered and/or time window evaluated for modeling a
given system. Multi-fidelity surrogate modeling aims to leverage less accurate,
lower-fidelity models that are computationally inexpensive in order to enhance
predictive accuracy when high-fidelity data are limited or scarce. However,
low-fidelity models, while often displaying important qualitative
spatio-temporal features, fail to accurately capture the onset of instability
and critical transients observed in the high-fidelity models, making them
impractical as surrogate models. To address this shortcoming, we present a new
data-driven strategy that combines dimensionality reduction with multi-fidelity
neural network surrogates. The key idea is to generate a spatial basis by
applying the classical proper orthogonal decomposition (POD) to high-fidelity
solution snapshots, and approximate the dynamics of the reduced states -
time-parameter-dependent expansion coefficients of the POD basis - using a
multi-fidelity long-short term memory (LSTM) network. By mapping low-fidelity
reduced states to their high-fidelity counterpart, the proposed reduced-order
surrogate model enables the efficient recovery of full solution fields over
time and parameter variations in a non-intrusive manner. The generality and
robustness of this method is demonstrated by a collection of parametrized,
time-dependent PDE problems where the low-fidelity model can be defined by
coarser meshes and/or time stepping, as well as by misspecified physical
features. Importantly, the onset of instabilities and transients are well
captured by this surrogate modeling technique.

应用多保真度替代建模方法，结合降维技术和神经网络代理，能有效地恢复高保真度数值模拟，捕捉系统的不稳定性和临界瞬变特征。