Physics-Informed Neural Networks for coupled rate-and-state friction and pore-pressure evolution

Summary Earthquake fault slip arises from nonlinear coupling among frictional evolution, elastic loading, and pore-pressure changes. When pore pressure evolves dynamically, the resulting hydro-mechanical rate-and-state models can be stiff and strongly coupled, making parameter inversion computationally demanding. Here we develop a physics-informed neural network (PINN) solver for a coupled spring–slider system that combines rate-and-state friction with pore-pressure/porosity evolution. The network approximates the time-dependent state variables and is trained by enforcing the governing differential equations together with initial conditions and, for inverse problems, observational constraints. To improve training stability, we employ adaptive inverse-residual weighting and a two-stage optimization schedule (Adam followed by L-BFGS). In forward simulations, PINN predictions closely match a Runge–Kutta reference solution across steady sliding and slow-slip transients, with normalized mean squared error below 0.08 and Pearson correlation coefficient above 0.975 for block velocity and frictional shear stress in the cases tested. In inverse experiments, the framework recovers the applied normal stress from noisy shear-stress observations; uncertainty increases with noise amplitude, but the ensemble mean remains stable, and at the highest noise level considered (q = 1) the inferred normal stress deviates by less than ~1% from the reference value. These results suggest that PINNs provide a differentiable alternative for forward modeling and parameter inversion in coupled hydro-mechanical rate-and-state fault models.