(Synthetic) Data-driven identification of governing equations of aseismic/seismic fault slip and frictional heterogeneities

Models of fault slip development generally consider interfacial strength to be frictional and deformation of the bounding medium to be elastic. The frictional strength is usually considered as sliding rate- and state-dependent. Their combination, elastic deformation due to differential slip and rate-state frictional strength, leads to nonlinear partial differential equations (PDEs) that govern the spatio-temporal evolution of slip. Here, we investigate how data on fault slip rate and stress** can directly discover the complex system (of PDEs) that governs aseismic slip development. **We first prepare (synthetic) data sets by numerically solving the forward problem of slip rate and fault stress evolution with models, such as a thin laterally deformable layer over a thick substrate. We now identify the variables, for example, slip rate or friction state variable, and** **use** nonlinearity identification algorithms to discover the governing PDE of the chosen variable.**

In particular, we use sparse identification of nonlinear dynamics algorithm (SINDy, Brunton et al., 2016) where we solve a regression problem, **Ax=y**. Here, **y** is the time derivative of the variable of interest, for example, slip rate. **A** is a large matrix (library) with all possible candidate functions that may appear in the slip rate evolution PDE. The entries in **x**, to be solved for, are coefficients corresponding to each library function in matrix **A**. We update **A** according to the solutions **x **so that **A**'s column space can span the dynamics we seek to find. To find the suitable column space for **A**,** **we encourage sparse solutions for **x**,** **suggesting that only a few columns in matrix **A **are dominant, leading to** **a parsimonious representation of the governing PDE.

We show that the algorithm successfully recovers the terms of the PDE governing fault slip and could also find the frictional parameter, for example, a/b, where a and b, respectively, are the magnitudes that control direct and evolution effects. Moreover, the algorithm can also determine whether the associated state variable evolves as aging- or slip-law types or their combination. Further, with the data set prepared from distinct initial conditions, we show that the SINDy can also determine the **problem parameterâ€™s spatial distribution (heterogeneities)** from fault slip rate and stress data.

01 Mar 2023

01 Mar 2023