We present a physics-informed machine-learning (PIML) approach for the approximation of slow invariant manifolds (SIMs) of singularly perturbed systems, providing functionals in an explicit form that facilitate the construction and numerical integration of reduced order models (ROMs). The proposed scheme solves a partial differential equation corresponding to the invariance equation (IE) within the Geometric Singular Perturbation Theory (GSPT) framework. For the solution of the IE, we used two neural network structures, namely feedforward neural networks (FNNs), and random projection neural networks (RPNNs), with symbolic differentiation for the computation of the gradients required for the learning process. The efficiency of our PIML method is assessed via three benchmark problems, namely the Michaelis-Menten, the target mediated drug disposition reaction mechanism, and the 3D Sel'kov model. We show that the proposed PIML scheme provides approximations, of equivalent or even higher accuracy, than those provided by other traditional GSPT-based methods, and importantly, for any practical purposes, it is not affected by the magnitude of the perturbation parameter. This is of particular importance, as there are many systems for which the gap between the fast and slow timescales is not that big, but still ROMs can be constructed. A comparison of the computational costs between symbolic, automatic and numerical approximation of the required derivatives in the learning process is also provided.

翻译：我们提出了一种基于物理信息的机器学习方法，用于逼近奇异摄动系统的慢不变流形，以显式函数形式提供泛函，从而促进降阶模型的构建与数值积分。该方案在几何奇异摄动理论框架下求解与不变方程对应的偏微分方程。为求解不变方程，我们采用了两种神经网络结构——前馈神经网络和随机投影神经网络，并利用符号微分计算学习过程中所需的梯度。通过三个基准问题（米氏-门滕动力学、靶标介导的药物处置反应机制及三维Sel'kov模型）评估了该物理信息机器学习方法的效率。研究表明，本文提出的方法能提供与传统几何奇异摄动理论方法精度相当甚至更高的逼近结果，且重要的是，在实际应用中其精度不受摄动参数大小的影响。这一特性尤为重要，因为许多系统的快慢时间尺度间隙虽不显著，但降阶模型仍可构建。此外，我们还比较了学习过程中所需导数的符号微分、自动微分与数值近似三种计算方式的成本。