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.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equations for flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equation for anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

Molecular dynamics simulations are employed to study the mechanical properties of nanoporous CuxZr1-x metallic glasses (MGs) with five different compositions, x = 0.28, 0.36, 0.50, 0.64, and 0.72, and porosity in the range 0.1 < ϕ < 0.7. Results from tensile loading simulations indicate a strong dependence of Young's modulus, E, and Ultimate Tensile Strength (UTS) on porosity and composition. By increasing the porosity from ϕ = 0.1 to ϕ = 0.7, the topology of the nanoporous MG shifts from closed cell to open-cell bicontinuous. The change in nanoporous topology enables a brittle-to-ductile transition in deformation and failure mechanisms from a single critical shear band to necking and rupture of ligaments. Genetic Programming (GP) is employed to find scaling laws for E and UTS as a function of porosity and composition. A comparison of the GP-derived scaling laws against existing relationships shows that the GP method is able to uncover expressions that can predict accurately both the values of E and UTS in the whole range of porosity and compositions considered.