Physics-Informed Deep Learning of Rate-and-State Fault Friction

TL;DR

This work develops a physics-informed neural network (PINN) framework to model elastodynamic wave propagation with rate-and-state friction on faults, enabling simultaneous forward and inverse problems on 1D and 2D strike-slip faults. A multi-network architecture solves the forward problem for displacement and a state/friction network to learn depth-dependent RSF parameters, with a 2D extension learning $\alpha(z)=a-b$ across depth. The authors verify the approach using the method of manufactured solutions, compare soft and hard enforcement of boundary conditions, and show that learning the friction depth profile via a dedicated network can recover $\alpha(z)$ accurately, with hard boundary enforcement often yielding better state and parameter estimates. This PINN framework provides a mesh-free path to integrate sparse surface observations with physics, offering a scalable tool for inferring subsurface friction properties and improving seismic hazard assessments.

Abstract

Direct observations of earthquake nucleation and propagation are few and yet the next decade will likely see an unprecedented increase in indirect, surface observations that must be integrated into modeling efforts. Machine learning (ML) excels in the presence of large data and is an actively growing field in seismology. However, not all ML methods incorporate rigorous physics, and purely data-driven models can predict physically unrealistic outcomes due to observational bias or extrapolation. Our work focuses on the recently emergent Physics-Informed Neural Network (PINN), which seamlessly integrates data while ensuring that model outcomes satisfy rigorous physical constraints. In this work we develop a multi-network PINN for both the forward problem as well as for direct inversion of nonlinear fault friction parameters, constrained by the physics of motion in the solid Earth, which have direct implications for assessing seismic hazard. We present the computational PINN framework for strike-slip faults in 1D and 2D subject to rate-and-state friction. Initial and boundary conditions define the data on which the PINN is trained. While the PINN is capable of approximating the solution to the governing equations to low-errors, our primary interest lies in the network's capacity to infer friction parameters during the training loop. We find that the network for the parameter inversion at the fault performs much better than the network for material displacements to which it is coupled. Additional training iterations and model tuning resolves this discrepancy, enabling a robust surrogate model for solving both forward and inverse problems relevant to seismic faulting.

Figures (6)

• Figure 1: A schematic of the PINN framework for solving the general boundary value problem from equation \ref{['eqn: ibvp(operator form)']}. Displacement approximation network $\mathcal{N}$ is trained on interior and boundary subdomains which are governed by operators $\mathcal{L}$ and $\mathcal{B}$, respectively.
• Figure 2: A schematic of the PINN framework for solving the 1D IBVP \ref{['eqn: 1Delastodynamic']}\ref{['eqn: 1D boundary conditions']} with rate and state friction defined by \ref{['eqn: 1D RNS fritction']} and \ref{['eqn: aging']}. Displacement network $\mathcal{N}$ and state network $\mathcal{N}^\psi$ are trained separately by defining two objective functions which are differentiated by red and blue nodes, respectively, in the final layer. At each training iteration, $\mathcal{N}$ is updated using component losses $MSE_\Omega, MSE_0,$ and $MSE_1$ while $\mathcal{N}^\psi$ is updated using just $MSE_\psi$ as an objective function.
• Figure 3: Comparison of results from 1D illustration showing the (a) displacement network approximation $\mathcal{N}$ with (b) manufactured solution $u^e$. Additionally, the (c) state approximation network $\mathcal{N}^\psi$ is plotted against the manufactured state $\psi^e$ along with their absolute error $|\mathcal{N}^\psi(t) - \psi^e(t)|$. Absolute displacement error was averaged over 1000 randomly sampled points and measured to be $|\mathcal{N}-u^e|_{\text{avg}}=1.57e-5$.
• Figure 4: Network diagram for enforcing the stress condition along the fault while also learning the depth-dependent friction parameter $\alpha$. During training, the $z$-coordinate of a training point is passed to both $\mathcal{N}$ and $\mathcal{N}^\alpha$.
• Figure 5: (a) 2D displacement plot for a PINN trained to solve the inverse problem using hard enforcement of initial conditions compared to (b) the manufactured displacements. (c)$L^2-$errors for displacement in space, time, and spacetime (along with $L^2-$errors for the friction parameter) are computed on a uniform grid using Simpson's rule as a quadrature. Errors are then recorded over several mesh refinements.