Experimental data is often comprised of variables measured independently, at different sampling rates (non-uniform ${\Delta}$t between successive measurements); and at a specific time point only a subset of all variables may be sampled. Approaches to identifying dynamical systems from such data typically use interpolation, imputation or subsampling to reorganize or modify the training data $\textit{prior}$ to learning. Partial physical knowledge may also be available $\textit{a priori}$ (accurately or approximately), and data-driven techniques can complement this knowledge. Here we exploit neural network architectures based on numerical integration methods and $\textit{a priori}$ physical knowledge to identify the right-hand side of the underlying governing differential equations. Iterates of such neural-network models allow for learning from data sampled at arbitrary time points $\textit{without}$ data modification. Importantly, we integrate the network with available partial physical knowledge in "physics informed gray-boxes"; this enables learning unknown kinetic rates or microbial growth functions while simultaneously estimating experimental parameters.

翻译：实验数据通常包含以不同采样率独立测量的变量（连续测量之间的时间间隔${\Delta}$t不统一）；并且在特定时间点，可能仅对全部变量中的一部分进行采样。从这类数据中辨识动态系统的方法通常在学习过程$\textit{之前}$，通过插值、归因或降采样来重组或修改训练数据。此外，部分物理知识也可能$\textit{先验}$存在（准确或近似），而数据驱动技术可以补充这些知识。本文利用基于数值积分方法和$\textit{先验}$物理知识的神经网络架构，来辨识底层控制微分方程的右侧表达式。此类神经网络模型的迭代过程允许从任意时间点采样的数据中进行学习，$\textit{无需}$对数据进行修改。关键在于，我们将网络与可用的部分物理知识相结合，形成"物理信息灰箱"；这使得我们能够学习未知动力学速率或微生物生长函数，同时同步估计实验参数。