The simulation of many complex phenomena in engineering and science requires
solving expensive, high-dimensional systems of partial differential equations
(PDEs). To circumvent this, reduced-order models (ROMs) have been developed to
speed up computations. However, when governing equations are unknown or
partially known, typically ROMs lack interpretability and reliability of the
predicted solutions.
In this work we present a data-driven, non-intrusive framework for building
ROMs where the latent variables and dynamics are identified in an interpretable
manner and uncertainty is quantified. Starting from a limited amount of
high-dimensional, noisy data the proposed framework constructs an efficient ROM
by leveraging variational autoencoders for dimensionality reduction along with
a newly introduced, variational version of sparse identification of nonlinear
dynamics (SINDy), which we refer to as Variational Identification of Nonlinear
Dynamics (VINDy).
In detail, the method consists of Variational Encoding of Noisy Inputs (VENI)
to identify the distribution of reduced coordinates. Simultaneously, we learn
the distribution of the coefficients of a pre-determined set of candidate
functions by VINDy. Once trained offline, the identified model can be queried
for new parameter instances and new initial conditions to compute the
corresponding full-time solutions. The probabilistic setup enables uncertainty
quantification as the online testing consists of Variational Inference
naturally providing Certainty Intervals (VICI). In this work we showcase the
effectiveness of the newly proposed VINDy method in identifying interpretable
and accurate dynamical system for the R\"ossler system with different noise
intensities and sources. Then the performance of the overall method - named
VENI, VINDy, VICI - is tested on PDE benchmarks including structural mechanics
and fluid dynamics.

使用数据驱动的框架，结合变分自动编码器和稀疏非线性动力学识别的变分版本，提出了一种解释性和可靠的降阶模型构建方法，并进行了不确定性量化。通过英文缩写 VENI、VINDy 和 VICI 展示了该方法在识别可解释和准确的动力学系统以及对结构力学和流体力学的性能测试时的有效性。

VENI, VINDy, VICI：具有不确定性量化的变分降阶建模框架

VENI, VINDy, VICI: a variational reduced-order modeling framework with  uncertainty quantification

In this work we analyze the effectiveness of the Sparse Identification of
Nonlinear Dynamics (SINDy) technique on three benchmark datasets for nonlinear
identification, to provide a better understanding of its suitability when
tackling real dynamical systems. While SINDy can be an appealing strategy for
pursuing physics-based learning, our analysis highlights difficulties in
dealing with unobserved states and non-smooth dynamics. Due to the ubiquity of
these features in real systems in general, and control applications in
particular, we complement our analysis with hands-on approaches to tackle these
issues in order to exploit SINDy also in these challenging contexts.

我们分析了 Sparse Identification of Nonlinear Dynamics (SINDy) 技术在三个非线性识别基准数据集上的有效性，以更好地了解其适用性，当解决真实动力系统时。虽然 SINDy 是一种追求基于物理的学习的吸引人策略，但我们的分析强调了处理未观测状态和非光滑动力学的困难。由于这些特征在一般的真实系统以及控制应用中无处不在，我们通过实践方法来解决这些问题，以便在这些具有挑战性的环境中利用 SINDy。

SINDy 与硬非线性和隐藏动力学之间的基准研究

SINDy vs Hard Nonlinearities and Hidden Dynamics: a Benchmarking Study

The discovery of governing differential equations from data is an open
frontier in machine learning. The sparse identification of nonlinear dynamics
(SINDy) \citep{brunton_discovering_2016} framework enables data-driven
discovery of interpretable models in the form of sparse, deterministic
governing laws. Recent works have sought to adapt this approach to the
stochastic setting, though these adaptations are severely hampered by the curse
of dimensionality. On the other hand, Bayesian-inspired deep learning methods
have achieved widespread success in high-dimensional probabilistic modeling via
computationally efficient approximate inference techniques, suggesting the use
of these techniques for efficient stochastic equation discovery. Here, we
introduce HyperSINDy, a framework for modeling stochastic dynamics via a deep
generative model of sparse governing equations whose parametric form is
discovered from data. HyperSINDy employs a variational encoder to approximate
the distribution of observed states and derivatives. A hypernetwork
\citep{ha_hypernetworks_2016} transforms samples from this distribution into
the coefficients of a differential equation whose sparse form is learned
simultaneously using a trainable binary mask \citep{louizos_learning_2018}.
Once trained, HyperSINDy generates stochastic dynamics via a differential
equation whose coefficients are driven by a Gaussian white noise. In
experiments, HyperSINDy accurately recovers ground truth stochastic governing
equations, with learned stochasticity scaling to match that of the data.
Finally, HyperSINDy provides uncertainty quantification that scales to
high-dimensional systems. Taken together, HyperSINDy offers a promising
framework for model discovery and uncertainty quantification in real-world
systems, integrating sparse equation discovery methods with advances in
statistical machine learning and deep generative modeling.

HyperSINDy 通过稀疏非线性动力学的识别、随机动力学的建模、变分编码器、不确定性量化等方面的进展，提供了一个有前途的框架，用于在实际系统中进行模型发现和不确定性量化。