This position paper takes a broad look at Physics-Enhanced Machine Learning
(PEML) -- also known as Scientific Machine Learning -- with particular focus to
those PEML strategies developed to tackle dynamical systems' challenges. The
need to go beyond Machine Learning (ML) strategies is driven by: (i) limited
volume of informative data, (ii) avoiding accurate-but-wrong predictions; (iii)
dealing with uncertainties; (iv) providing Explainable and Interpretable
inferences. A general definition of PEML is provided by considering four
physics and domain knowledge biases, and three broad groups of PEML approaches
are discussed: physics-guided, physics-encoded and physics-informed. The
advantages and challenges in developing PEML strategies for guiding
high-consequence decision making in engineering applications involving complex
dynamical systems, are presented.

本篇综述研究物理增强的机器学习 (PEML) - 也被称为科学机器学习 - 特别关注开发用于解决动力系统挑战的 PEML 策略，讨论了 PEML 方法的三个广泛分类 (物理指导、物理编码和物理信息) 以及在涉及复杂动力系统的工程应用中开发 PEML 策略用于指导高风险决策的优势和挑战。

物理增强机器学习：动力系统研究的一个立场文件

Physics-Enhanced Machine Learning: a position paper for dynamical  systems investigations

The intersection of physics and machine learning has given rise to a paradigm
that we refer to here as physics-enhanced machine learning (PEML), aiming to
improve the capabilities and reduce the individual shortcomings of data- or
physics-only methods. In this paper, the spectrum of physics-enhanced machine
learning methods, expressed across the defining axes of physics and data, is
discussed by engaging in a comprehensive exploration of its characteristics,
usage, and motivations. In doing so, this paper offers a survey of recent
applications and developments of PEML techniques, revealing the potency of PEML
in addressing complex challenges. We further demonstrate application of select
such schemes on the simple working example of a single-degree-of-freedom
Duffing oscillator, which allows to highlight the individual characteristics
and motivations of different `genres' of PEML approaches. To promote
collaboration and transparency, and to provide practical examples for the
reader, the code of these working examples is provided alongside this paper. As
a foundational contribution, this paper underscores the significance of PEML in
pushing the boundaries of scientific and engineering research, underpinned by
the synergy of physical insights and machine learning capabilities.

本文讨论了物理增强机器学习（PEML）的特点、用途和动机，并通过对 PEML 技术的最近应用和发展的全面探索，展示了 PEML 在解决复杂挑战中的潜力。通过对单自由度 Duffing 振子的简单工作示例的应用，突出了不同 PEML 方法的个体特征和动机。提供这些工作示例的代码，以促进合作和透明度，并为读者提供实用的示例。作为基础性贡献，本文强调了物理洞察力和机器学习能力的协同作用，在推动科学和工程研究的边界上的重要性。