# Some of the variables, some of the parameters, some of the times, with some physics known: Identification with partial information

Saurabh Malani, Tom S. Bertalan, Tianqi Cui, Jos L. Avalos, Michael Betenbaugh, and Ioannis G. Kevrekidis. *Computers & Chemicial Engineering*, 178 (2023). https://doi.org/10.1016/j.compchemeng.2023.108343

Experimental data is often comprised of variables measured independently, at different sampling rates (non-uniform 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 prior to learning. Partial physical knowledge may also be available 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 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 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.