Finding governing PDEs of quasistatic fault slip and basal slip evolution from (synthetic) slip rate and shear traction data.

Mechanical models of slip development on geological faults and basal slip development in landslide or ice-sheets generally consider interfacial strength to be frictional and deformation of the surrounding 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 (synthetic) data on fault slip rate and traction can find the system of PDEs that governs fault slip development during the aseismic rupture phase and the slip instability phase. We first prepare (synthetic) data sets by numerically solving the forward problem of slip rate and fault shear stress evolution during a seismic cycle. We now identify the physical variables, for example, slip rate or frictional state variable, and apply nonlinearity identification algorithms within different time durations. We show that the nonlinearity identification algorithms can find the terms of the PDE that governs the slip rate evolution during the aseismic rupture phase and subsequent instability phase.

In particular, we use nonlinear dynamics identification algorithms (e.g., 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 PDE governing quasi-static fault slip and basal slip evolution. Additionally, we 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 nonlinear dynamics identification algorithm can also determine the problem parameters’ spatial distributions (heterogeneities) from fault slip rate and shear stress data.