This paper explores the prediction of the dynamics of piecewise smooth maps
using various deep learning models. We have shown various novel ways of
predicting the dynamics of piecewise smooth maps using deep learning models.
Moreover, we have used machine learning models such as Decision Tree
Classifier, Logistic Regression, K-Nearest Neighbor, Random Forest, and Support
Vector Machine for predicting the border collision bifurcation in the 1D normal
form map and the 1D tent map. Further, we classified the regular and chaotic
behaviour of the 1D tent map and the 2D Lozi map using deep learning models
like Convolutional Neural Network (CNN), ResNet50, and ConvLSTM via cobweb
diagram and phase portraits. We also classified the chaotic and hyperchaotic
behaviour of the 3D piecewise smooth map using deep learning models such as the
Feed Forward Neural Network (FNN), Long Short-Term Memory (LSTM), and Recurrent
Neural Network (RNN). Finally, deep learning models such as Long Short-Term
Memory (LSTM) and Recurrent Neural Network (RNN) are used for reconstructing
the two parametric charts of 2D border collision bifurcation normal form map.

本研究探讨了使用各种深度学习模型来预测分段平滑映射的动力学。通过使用深度学习模型，我们展示了预测分段平滑映射动力学的各种新颖方法。我们还使用机器学习模型如决策树分类器、逻辑回归、K 近邻、随机森林以及支持向量机来预测一维正规形式映射和一维帐篷映射中的边界碰撞分叉。此外，我们通过混沌图和相图，使用卷积神经网络（CNN）、ResNet50 和 ConvLSTM 来对一维帐篷映射和二维 Lozi 映射的规则和混沌行为进行分类。我们还使用前馈神经网络（FNN）、长短期记忆（LSTM）和循环神经网络（RNN）等深度学习模型来对三维分段平滑映射的混沌和超混沌行为进行分类。最后，我们使用长短期记忆（LSTM）和循环神经网络（RNN）来重建二维边界碰撞分叉正规形式映射的两个参数图。

深度学习预测和分类分段光滑映射的动态行为

Deep Learning for Prediction and Classifying the Dynamical behaviour of  Piecewise Smooth Maps

While the popularity of physics-informed neural networks (PINNs) is steadily
rising, to this date PINNs have not been successful in simulating dynamical
systems whose solution exhibits multi-scale, chaotic or turbulent behavior. In
this work we attribute this shortcoming to the inability of existing PINNs
formulations to respect the spatio-temporal causal structure that is inherent
to the evolution of physical systems. We argue that this is a fundamental
limitation and a key source of error that can ultimately steer PINN models to
converge towards erroneous solutions. We address this pathology by proposing a
simple re-formulation of PINNs loss functions that can explicitly account for
physical causality during model training. We demonstrate that this simple
modification alone is enough to introduce significant accuracy improvements, as
well as a practical quantitative mechanism for assessing the convergence of a
PINNs model. We provide state-of-the-art numerical results across a series of
benchmarks for which existing PINNs formulations fail, including the chaotic
Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the
Navier-Stokes equations in the turbulent regime. To the best of our knowledge,
this is the first time that PINNs have been successful in simulating such
systems, introducing new opportunities for their applicability to problems of
industrial complexity.

本文提出一种简单的 PINNs 卷积方法，可显式考虑物理因果关系，在多尺度、混沌或湍流行为的动力学系统中实现更高的准确性和可靠性。

尊重因果关系即可训练物理学感知神经网络

Respecting causality is all you need for training physics-informed  neural networks

Learning the governing equations in dynamical systems from time-varying
measurements is of great interest across different scientific fields. This task
becomes prohibitive when such data is moreover highly corrupted, for example,
due to the recording mechanism failing over unknown intervals of time. When the
underlying system exhibits chaotic behavior, such as sensitivity to initial
conditions, it is crucial to recover the governing equations with high
precision. In this work, we consider continuous time dynamical systems $\dot{x}
= f(x)$ where each component of $f: \mathbb{R}^{d} \rightarrow \mathbb{R}^d$ is
a multivariate polynomial of maximal degree $p$; we aim to identify $f$ exactly
from possibly highly corrupted measurements $x(t_1), x(t_2), \dots, x(t_m)$. As
our main theoretical result, we show that if the system is sufficiently ergodic
that this data satisfies a strong central limit theorem (as is known to hold
for chaotic Lorenz systems), then the governing equations $f$ can be exactly
recovered as the solution to an $\ell_1$ minimization problem -- even if a
large percentage of the data is corrupted by outliers. Numerically, we apply
the alternating minimization method to solve the corresponding constrained
optimization problem. Through several examples of 3D chaotic systems and higher
dimensional hyperchaotic systems, we illustrate the power, generality, and
efficiency of the algorithm for recovering governing equations from noisy and
highly corrupted measurement data.

通过混有噪声和异常值得动力系统测量数据，我们提出了一种使用多项式恢复其控制方程的算法，该算法能够适用于混沌系统中通过强中心极限定理满足的数据，并进行完美的精度恢复。该算法使用交替最小化进行实现，并在多个 3D 混沌系统和更高维超混沌系统实例上表现出了强大的通用性和高效性。