1:45 PM - 3:15 PM
[SCG55-P13] Physics-Informed Neural Networks for modeling slow slip events in a spring-slider system with a rate and state friction law
Keywords:PINNs, spring-slider model, earthquake cycle simulation, rate and state dependent friction law, slow slip events
Frictional properties on subducting plates determine spatio-temporal evolutions of slip. Therefore, understanding the frictional properties is important for predicting interplate earthquakes. Fault slips can be simulated by partial differential equations (PDEs) consisting of equation of motions and a rate and state dependent friction law, which is empirically derived from laboratory experiments (Ruina, 1983). Estimation of frictional parameters has been studied by data assimilation (Kano et al, 2015, 2020, Hirahara and Nishikiori 2019) integrating the physical model and observed slip data from GNSS.
Recent advances in machine learning provide a new method to solve the PDEs and to decide the controlling parameters of PDEs from the data. In the Physics-Informed Neural Networks (PINNs) approach, we construct neural networks that can solve the physics-based equations by minimizing the loss function which involves the differential equations and initial / boundary conditions (Raissi et al., 2019). This approach has been recently adopted in many research fields because it not only provides the mesh-free framework for forward problems but also easily obtains solutions for inverse problems. In seismology, PINNs has been applied to various problems including travel time calculation (Smith et al., 2021a), hypocenter inversion (Smith et al., 2021b), full-waveform inversion (Rasht-Behesht et al., 2022), seismic tomography (Agata et al., 2023) and modeling the crustal deformation (Okazaki et al., 2022).
In this study, we applied PINNs to the simulation of slip evolution on faults. We adopted a spring-slider model (Yoshida and Kato, 2003), which combines quasi-dynamic equations of motion (Rice, 1993) and rate and state friction law. Because of the strong non-linearity of fast earthquakes, the target is slow slip events (SSEs) in this study. The purpose of this study is to (i) simulate the slip rate evolutions of SSEs as the forward problem and (ii) simultaneously estimate the slip rate evolutions of SSEs and frictional parameters on faults from the slip rate data generated by PINNs as the inverse problem.
For solving the forward problem (i), we define the loss function as the weighted sum of the misfit of different equations and the misfit of initial conditions. By assigning the appropriate weight, we successfully reproduced SSEs. In the training of the neural networks, we need to set the collocation points where the misfit of differential equations is calculated. Unlike the time-adaptive Runge-Kutta approach that is usually used in solving these equations, PINNs can solve the equation even with equidistant collocation points. This indicates the high interpolation ability of PINNs.
One of the defects of PINNs is ignoring the temporal causal structure, though causality is important in non-linear problems such as earthquakes. To overcome this problem, we applied the Causal-PINNs (Wang et al., 2022) to the same problem. We found that the Causal-PINNs obtained a similar simulation result with faster calculation speed than original PINNs.
For solving the inverse problem (ii), we add the misfit term between the observed and calculated slip velocity data to the loss function. We optimize the frictional parameters a, a-b, and dc in addition to the neural network parameters. In this optimization, we define the logarithm of frictional parameters, α, β, and γ as a = eα, a – b = - eβ and dc = eγ and optimize these parameters. This means we use the prior information that a – b is negative and allows to search the large range of values. As a result, all frictional parameters were optimized from the synthetic data with observation noises.
These results imply that the PINNs approach is effective in earthquake cycle simulation and frictional parameter estimation.
Recent advances in machine learning provide a new method to solve the PDEs and to decide the controlling parameters of PDEs from the data. In the Physics-Informed Neural Networks (PINNs) approach, we construct neural networks that can solve the physics-based equations by minimizing the loss function which involves the differential equations and initial / boundary conditions (Raissi et al., 2019). This approach has been recently adopted in many research fields because it not only provides the mesh-free framework for forward problems but also easily obtains solutions for inverse problems. In seismology, PINNs has been applied to various problems including travel time calculation (Smith et al., 2021a), hypocenter inversion (Smith et al., 2021b), full-waveform inversion (Rasht-Behesht et al., 2022), seismic tomography (Agata et al., 2023) and modeling the crustal deformation (Okazaki et al., 2022).
In this study, we applied PINNs to the simulation of slip evolution on faults. We adopted a spring-slider model (Yoshida and Kato, 2003), which combines quasi-dynamic equations of motion (Rice, 1993) and rate and state friction law. Because of the strong non-linearity of fast earthquakes, the target is slow slip events (SSEs) in this study. The purpose of this study is to (i) simulate the slip rate evolutions of SSEs as the forward problem and (ii) simultaneously estimate the slip rate evolutions of SSEs and frictional parameters on faults from the slip rate data generated by PINNs as the inverse problem.
For solving the forward problem (i), we define the loss function as the weighted sum of the misfit of different equations and the misfit of initial conditions. By assigning the appropriate weight, we successfully reproduced SSEs. In the training of the neural networks, we need to set the collocation points where the misfit of differential equations is calculated. Unlike the time-adaptive Runge-Kutta approach that is usually used in solving these equations, PINNs can solve the equation even with equidistant collocation points. This indicates the high interpolation ability of PINNs.
One of the defects of PINNs is ignoring the temporal causal structure, though causality is important in non-linear problems such as earthquakes. To overcome this problem, we applied the Causal-PINNs (Wang et al., 2022) to the same problem. We found that the Causal-PINNs obtained a similar simulation result with faster calculation speed than original PINNs.
For solving the inverse problem (ii), we add the misfit term between the observed and calculated slip velocity data to the loss function. We optimize the frictional parameters a, a-b, and dc in addition to the neural network parameters. In this optimization, we define the logarithm of frictional parameters, α, β, and γ as a = eα, a – b = - eβ and dc = eγ and optimize these parameters. This means we use the prior information that a – b is negative and allows to search the large range of values. As a result, all frictional parameters were optimized from the synthetic data with observation noises.
These results imply that the PINNs approach is effective in earthquake cycle simulation and frictional parameter estimation.