<p>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<strong>&#160;can find the system of PDEs that governs fault slip development during the aseismic rupture phase and the slip instability phase.&#160;</strong>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<strong>&#160;</strong>nonlinearity identification algorithms&#160;within different time durations.&#160;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.</p><p>In particular, we use nonlinear dynamics identification algorithms (e.g., SINDy, Brunton et al., 2016) where we solve a regression problem, <strong>Ax=y.</strong> Here,&#160;<strong>y</strong>&#160;is the time derivative of the variable of interest, for example, slip rate.&#160;<strong>A</strong>&#160;is a large matrix (library) with all possible candidate functions that may appear in the slip rate evolution PDE. The entries in&#160;<strong>x</strong>, to be solved for, are coefficients corresponding to each library function in matrix&#160;<strong>A</strong>. We update&#160;<strong>A</strong>&#160;according to the solutions&#160;<strong>x&#160;</strong>so that&#160;<strong>A</strong>'s column space can span the dynamics we seek to find. To find the suitable column space for&#160;<strong>A,&#160;</strong>we encourage sparse solutions for&#160;<strong>x,&#160;</strong>suggesting that only a few columns in matrix&#160;<strong>A&#160;</strong>are dominant, leading to<strong>&#160;</strong>a parsimonious representation of the governing PDE.&#160;</p><p>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 <strong>problem parameters&#8217; spatial distributions (heterogeneities)</strong>&#160;from fault slip rate and shear stress data.&#160;</p>
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 parameter’s spatial distribution (heterogeneities) from fault slip rate and stress data.
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 parameter’s spatial distribution (heterogeneities) from fault slip rate and stress data.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.