Optimal Experimental Design for Universal Differential Equations

Complex dynamic systems are typically either modeled using expert knowledge in the form of differential equations or via data-driven universal approximation models such as artificial neural networks (ANN). While the first approach has advantages with respect to interpretability, transparency, data-efficiency, and extrapolation, the second approach is able to learn completely unknown functional relations from data and may result in models that can be evaluated more efficiently. To combine the complementary advantages, universal differential equations (UDE) have been suggested. They replace unknown terms in the differential equations with ANN. Such hybrid models allow to both encode prior domain knowledge, such as first principles, and to learn unknown mechanisms from data. Often, data for the training of UDE can only be obtained via costly experiments. We consider optimal experimental design (OED) for planning of experiments and generating data needed to train UDE. The number of weights in the embedded ANN usually leads to an overfitting of the regression problem. To make the OED problem tractable for optimization, we propose and compare dimension reduction methods that are based on lumping of weights and singular value decomposition of the Fisher information matrix (FIM), respectively. They result in lower-dimensional variational differential equations, which are easier to solve and yield regular FIM. Our numerical results showcase the advantages of OED for UDE, such as increased data-efficiency and better extrapolation properties.