Data-driven identification of partial differential equations for multi-physics systems using stochastic optimization

We develop a stochastic optimization framework to identify governing equations in multi-physics systems. The proposed approach discovers partial differential equations (PDEs) by combining user's prior knowledge of the underlying physics of a target system and its observed data. The technique relies on evolutionary processes to randomly generate PDEs and stochastically optimize their structure and coefficients to the data by exploring the infinite model space. Furthermore, the method captures the spatiotemporal dynamics of physical system by direct evaluation of the candidate PDEs under physical constraints. To achieve significant computational speedup, the proposed stochastic optimization method relies on a series of novel modifications. These consist of the incorporation of a multi-purpose loss function into the parallel fitness test and bloat control techniques into the evolutionary processes. As such, these innovations lead to a significant improvement of the effectiveness and computational efficiency in identifying PDEs from given data. We demonstrate the applicability of methodology in two illustrative examples: the nonlinear Burgers' equation and the linear/nonlinear advection-dispersion equation. The exact PDEs are successfully identified, even with significant data noise, and captures the underlying physics of the target system. Through series of simulations, we assess the accuracy, robustness, and limitations of the proposed approach. In particular, we show the impact of key dimensionless groups (that accounts for the competition between various physical phenomena) in controlling the accuracy of the identification process. This work shows that developed identification method is a promising effective and robust gray box modeling tool for identifying PDEs.