标题： 基于物理的机器学习：一个数学框架及其在时间序列预测中的应用 显示英文标题

标题： Physics-informed machine learning: A mathematical framework with applications to time series forecasting

摘要： 物理信息机器学习（PIML）是一种新兴的框架，它将物理知识整合到机器学习模型中。 这种物理先验通常以偏微分方程（PDE）系统的形式出现，回归函数必须满足该系统。 在本论文的第一部分，我们分析了PIML方法的统计特性。 特别是，我们从逼近性、一致性、过拟合和收敛性等方面研究了物理信息神经网络（PINNs）的特性。 然后，我们展示了如何将PIML问题表述为核方法，从而可以应用核岭回归的工具来更好地理解其行为。 此外，我们利用这种核公式开发了新颖的物理信息算法，并在GPU上高效实现它们。 第二部分探讨了在非常时期预测能源信号的工业应用。 我们介绍了电动汽车充电占用的智能交通挑战的结果，并研究了移动性对电力需求的影响。 最后，我们介绍了一个物理约束框架，用于设计和强制时间序列中的约束，并将其应用于不同国家的负荷预测和旅游业预测。 显示英文摘要

摘要： Physics-informed machine learning (PIML) is an emerging framework that integrates physical knowledge into machine learning models. This physical prior often takes the form of a partial differential equation (PDE) system that the regression function must satisfy. In the first part of this dissertation, we analyze the statistical properties of PIML methods. In particular, we study the properties of physics-informed neural networks (PINNs) in terms of approximation, consistency, overfitting, and convergence. We then show how PIML problems can be framed as kernel methods, making it possible to apply the tools of kernel ridge regression to better understand their behavior. In addition, we use this kernel formulation to develop novel physics-informed algorithms and implement them efficiently on GPUs. The second part explores industrial applications in forecasting energy signals during atypical periods. We present results from the Smarter Mobility challenge on electric vehicle charging occupancy and examine the impact of mobility on electricity demand. Finally, we introduce a physics-constrained framework for designing and enforcing constraints in time series, applying it to load forecasting and tourism forecasting in various countries.