In many scientific and engineering (e.g., physical, biochemical, medical) practices, data generated through expensive experiments or large-scale simulations, are often sparse and noisy. Physics-informed neural network (PINN) incorporates physical information and knowledge into network topology or computational processes as model priors, with the unique advantage of achieving strong generalization with small data. This study aims to investigate the performance characteristics of the soft-constrained PINN method to solving typical linear and nonlinear ordinary differential equations (ODEs) such as primer, Van der Pol and Duffing oscillators, especially the effectiveness, efficiency, and robustness to noise with minimal data. It is verified that the soft-constrained PINN significantly reduces the need for labeled data. With the aid of appropriate collocation points no need to be labeled, it can predict and also extrapolate with minimal data. First-order and second-order ODEs, no matter linear or nonlinear oscillators, require only one and two training data (containing initial values) respectively, just like classical analytic or Runge-Kutta methods, and with equivalent precision and comparable efficiency (fast training in seconds for scalar ODEs). Furthermore, it can conveniently impose a physical law (e.g., conservation of energy) constraint by adding a regularization term to the total loss function, improving the performance to deal with various complexities such as nonlinearity like Duffing. The DeepXDE-based PINN implementation is light code and can be efficiently trained on both GPU and CPU platforms. The mathematical and computational framework of this alternative and feasible PINN method to ODEs, can be easily extended to PDEs, etc., and is becoming a favorable catalyst for the era of Digital Twins.

翻译：在许多科学与工程（如物理、生化、医疗）实践中，通过昂贵实验或大规模仿真产生的数据往往稀疏且含有噪声。物理信息神经网络（PINN）将物理信息与知识作为模型先验融入网络拓扑或计算过程，具有以小样本实现强泛化的独特优势。本研究旨在探究软约束PINN方法在求解典型线性和非线性常微分方程（如初值、范德波尔与杜芬振荡器）时的性能特征，特别是在极小数据量下对噪声的有效性、效率与鲁棒性。研究验证了软约束PINN能显著降低对标注数据的需求。借助无需标注的合适配置点，该方法能以极小数据实现预测与外推。无论线性或非线性振荡器，一阶与二阶常微分方程分别仅需一个和两个训练数据（含初始值），即可达到与传统解析法或龙格-库塔法相当的精度与可比效率（标量常微分方程秒级快速训练）。此外，通过向总损失函数添加正则化项，该方法可便捷地施加物理定律（如能量守恒）约束，从而提升处理杜芬方程等非线性复杂问题的性能。基于DeepXDE的PINN实现代码轻量，可在GPU与CPU平台上高效训练。这种面向常微分方程的替代性可行PINN数学计算框架，可轻松拓展至偏微分方程等领域，正成为数字孪生时代的重要催化剂。