Table of Links

Abstract and 1. Context and Motivation

2. Physics-Informed Deep Learning Framework
3. Learning Problems for Earthquakes on Rate-and-State Faults
4. 2D Verification, Validation and Applications
5. Summary and Future Work and References

5. Summary and Future Work

We have presented a computational framework for physics-informed neural networks (PINNs) for solving the elastodynamic wave equation with a rate-and-state frictional fault boundary in both 1D and 2D. We consider both forward and inverse problems, with the latter obtained by extending to a multi-network architecture in order to learn depth-dependent friction parameters alongside deformations in the Earth’s crust. We verified the computational framework by applying the method of manufactured solutions to probe various error measurements. We show that in general, hard enforcement of boundary conditions result in trained networks that better approximate displacements and friction parameters but tend to be worse at minimizing component loss functions when compared to soft enforcement. We found that a PINN defined by hard enforcement of initial conditions produces reasonable approximations for displacements and the desired friction parameter distribution. Though the network is meshfree, we show that successive mesh refinements of the quadrature approximation to the L 2 -error yield near constant values, suggesting that the PINN provides a reasonable approximation to the solution even when evaluated on increasingly finer grids. Moreover, although the network requires sufficient training iterations to properly learn displacements, the desired friction parameter is learned within the first couple of iterations. This suggests that PINNs may be a highly effective tool in inferring subsurface friction properties along faults, constrained by both physics and observational surface data.

While the PINN is shown to perform well when learning the state variable in 1D, and inferring depth-dependency of RSF parameter a − b at steady-state in 2D, we plan to explore the capabilities of the 2D framework with non-steadystate-evolution. This extension requires additional networks in the 2D setting in order to approximate the state variable, and two inference networks to capture the empirical parameters a and b (which become separated across governing equations), and/or networks to learn other frictional parameters, such as Dc, whose scaling from laboratory values to actual fault zones is the subject of many studies [e.g. 37]. Approaching such a problem may be aided by a better understanding of the PINN dependence on problem configuration as well as network architecture. Additionally, it would be worthwhile to investigate methods which hybridize PINNs with traditional numerical methods similar to the discrete time PINN in Raissi et al. [43]. And finally, with a PINN solution that can handle learning full rate-and-state fault friction in 2D (and eventually 3D), we would be ready to compare model outcomes against community benchmark problems concerning dynamic rupture simulations [20] and sequences of earthquakes and aseismic slip [12].

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