Pareto-based Optimization of Sparse Dynamical Systems

Sparse data-driven approaches enable the approximation of governing laws of physical processes with parsimonious equations. While a great effort over the last decade has been made in this field, data-driven approaches generally rely on the paradigm of imposing a fixed base of library functions. In order to promote sparsity, finding the optimal set of basis functions is a necessary condition but a challenging task to guess in advance.Here, we propose an alternative approach which consists of optimizing the very library of functions while imposing sparsity. The robustness of our results is not only evaluated by the quality of the fit of the discovered model, but also by the statistical distribution of the residuals with respect to the original noise in the data. In order to avoid to choose one metric over the other, we rather rely on a multi-objective genetic algorithm (NSGA-II) for systematically generating a subset of optimal models sorted in a Pareto front. We illustrate how this method can be used as a tool to derive microkinetic equations from experimental data, and as a kernel approach for design of experiments.