Physics-informed neural networks (PINNs) have proven a suitable mathematical
scaffold for solving inverse ordinary (ODE) and partial differential equations
(PDE). Typical inverse PINNs are formulated as soft-constrained multi-objective
optimization problems with several hyperparameters. In this work, we
demonstrate that inverse PINNs can be framed in terms of maximum-likelihood
estimators (MLE) to allow explicit error propagation from interpolation to the
physical model space through Taylor expansion, without the need of
hyperparameter tuning. We explore its application to high-dimensional coupled
ODEs constrained by differential algebraic equations that are common in
transient chemical and biological kinetics. Furthermore, we show that
singular-value decomposition (SVD) of the ODE coupling matrices (reaction
stoichiometry matrix) provides reduced uncorrelated subspaces in which PINNs
solutions can be represented and over which residuals can be projected.
Finally, SVD bases serve as preconditioners for the inversion of covariance
matrices in this hyperparameter-free robust application of MLE to
``kinetics-informed neural networks''.

本文提出了一种基于最大似然估计的反向物理知识神经网络方法，解决高维背景物理时空方程的拟合问题，并利用奇异值分解作为预处理器进行协方差矩阵求逆，从而省去了超参数调整的必要性。