We present a physics-informed neural network (PINN) approach for the discovery of slow invariant manifolds (SIMs), for the most general class of fast/slow dynamical systems of ODEs. In contrast to other machine learning (ML) approaches that construct reduced order black box surrogate models using simple regression, and/or require a priori knowledge of the fast and slow variables, our approach, simultaneously decomposes the vector field into fast and slow components and provides a functional of the underlying SIM in a closed form. The decomposition is achieved by finding a transformation of the state variables to the fast and slow ones, which enables the derivation of an explicit, in terms of fast variables, SIM functional. The latter is obtained by solving a PDE corresponding to the invariance equation within the Geometric Singular Perturbation Theory (GSPT) using a single-layer feedforward neural network with symbolic differentiation. The performance of the proposed physics-informed ML framework is assessed via three benchmark problems: the Michaelis-Menten, the target mediated drug disposition (TMDD) reaction model and a fully competitive substrate-inhibitor(fCSI) mechanism. We also provide a comparison with other GPST methods, namely the quasi steady state approximation (QSSA), the partial equilibrium approximation (PEA) and CSP with one and two iterations. We show that the proposed PINN scheme provides SIM approximations, of equivalent or even higher accuracy, than those provided by QSSA, PEA and CSP, especially close to the boundaries of the underlying SIMs.

翻译：本文提出了一种基于物理信息神经网络（PINN）的方法，用于发现最一般类别的快/慢常微分方程动力系统中的慢不变流形（SIM）。与其它机器学习（ML）方法（如通过简单回归构建降阶黑箱替代模型，和/或需要预知快慢变量）不同，我们的方法同时将向量场分解为快、慢分量，并提供底层SIM的封闭形式函数。该分解通过寻找状态变量到快慢变量的变换实现，从而推导出关于快变量的显式SIM函数。后者通过几何奇异摄动理论（GSPT）中的不变方程对应的偏微分方程求解获得，并采用单层前馈神经网络结合符号微分实现。我们通过三个基准问题评估了所提出的物理信息ML框架的性能：米氏-门滕动力学模型、靶向介导药物处置（TMDD）反应模型，以及完全竞争性底物-抑制剂（fCSI）机制。此外，我们还与其它GSPT方法进行了比较，包括准稳态近似（QSSA）、部分平衡近似（PEA）以及单次和两次迭代CSP。结果表明，所提出的PINN方案提供的SIM近似精度与QSSA、PEA和CSP相当甚至更高，尤其在底层SIM边界附近优势显著。