Data-driven modeling can suffer from a constant demand for data, leading to
reduced accuracy and impractical for engineering applications due to the high
cost and scarcity of information. To address this challenge, we propose a
progressive reduced order modeling framework that minimizes data cravings and
enhances data-driven modeling's practicality. Our approach selectively
transfers knowledge from previously trained models through gates, similar to
how humans selectively use valuable knowledge while ignoring unuseful
information. By filtering relevant information from previous models, we can
create a surrogate model with minimal turnaround time and a smaller training
set that can still achieve high accuracy. We have tested our framework in
several cases, including transport in porous media, gravity-driven flow, and
finite deformation in hyperelastic materials. Our results illustrate that
retaining information from previous models and utilizing a valuable portion of
that knowledge can significantly improve the accuracy of the current model. We
have demonstrated the importance of progressive knowledge transfer and its
impact on model accuracy with reduced training samples. For instance, our
framework with four parent models outperforms the no-parent counterpart trained
on data nine times larger. Our research unlocks data-driven modeling's
potential for practical engineering applications by mitigating the data
scarcity issue. Our proposed framework is a significant step toward more
efficient and cost-effective data-driven modeling, fostering advancements
across various fields.

通过减少数据需求并提高数据驱动模型的实际性，我们提出了一种渐进降阶建模框架，通过选择性地从之前的训练模型中转移有价值的知识，从而创建一个具有高准确性的代理模型。我们的研究通过缓解数据稀缺问题，解锁了数据驱动建模在实际工程应用方面的潜力，为各个领域的进步提供了强有力的支持。

渐进降阶建模：以选择性知识传输强化数据驱动建模

Progressive reduced order modeling: empowering data-driven modeling with  selective knowledge transfer

In this article, we present a data-driven method for parametric models with
noisy observation data. Gaussian process regression based reduced order
modeling (GPR-based ROM) can realize fast online predictions without using
equations in the offline stage. However, GPR-based ROM does not perform well
for complex systems since POD projection are naturally linear. Conditional
variational autoencoder (CVAE) can address this issue via nonlinear neural
networks but it has more model complexity, which poses challenges for training
and tuning hyperparameters. To this end, we propose a framework of CVAE with
Gaussian process regression recognition (CVAE-GPRR). The proposed method
consists of a recognition model and a likelihood model. In the recognition
model, we first extract low-dimensional features from data by POD to filter the
redundant information with high frequency. And then a non-parametric model GPR
is used to learn the map from parameters to POD latent variables, which can
also alleviate the impact of noise. CVAE-GPRR can achieve the similar accuracy
to CVAE but with fewer parameters. In the likelihood model, neural networks are
used to reconstruct data. Besides the samples of POD latent variables and input
parameters, physical variables are also added as the inputs to make predictions
in the whole physical space. This can not be achieved by either GPR-based ROM
or CVAE. Moreover, the numerical results show that CVAE-GPRR may alleviate the
overfitting issue in CVAE.

本文提出了一种基于数据驱动的方法，结合高斯过程回归和条件变分自编码器 (CVAE-GPRR)，用于参数模型，在实现快速在线预测的同时避免模型复杂性和噪声的影响。

使用高斯过程回归识别的条件变分自编码器用于参数模型

Conditional variational autoencoder with Gaussian process regression  recognition for parametric models

Data-driven reduced order modeling of chaotic dynamics can result in systems
that either dissipate or diverge catastrophically. Leveraging non-linear
dimensionality reduction of autoencoders and the freedom of non-linear operator
inference with neural-networks, we aim to solve this problem by imposing a
synthetic constraint in the reduced order space. The synthetic constraint
allows our reduced order model both the freedom to remain fully non-linear and
highly unstable while preventing divergence. We illustrate the methodology with
the classical 40-variable Lorenz '96 equations, showing that our methodology is
capable of producing medium-to-long range forecasts with lower error using less
data.

使用自编码器的非线性降维和神经网络的非线性操作推断，通过在降维空间中引入合成约束，解决了混沌动力学的降阶建模问题，使模型既能够保持完全非线性和高度不稳定，又能防止发散，在经典的 Lorenz 方程中进行了演示，并表明我们的方法可以使用更少的数据产生较低误差的中长程预测。