A Physically Constrained KPP–Rate-and-State Reaction–Diffusion Framework for Stable Large-Scale Modeling of Stress Evolution

The emergence of large-scale models and machine learning has transformed the modeling of complex nonlinear systems, such as postseismic stress evolution. However, purely data-driven approaches often lack interpretability and numerical stability, leading to physically inconsistent long-term predictions. This study addresses these limitations by introducing a coupled Kolmogorov–Petrovsky–Piskunov–Rate-and-State (KPP–RS) reaction–diffusion system as a rigorous physical prior for large-scale modeling of stress-driven dynamics. Using analytic semigroup theory and Banach’s fixed-point theorem, we establish the global existence and uniqueness of solutions, ensuring that the governing dynamics are mathematically well posed—a necessary prerequisite for stable learning-based frameworks. We further prove the global dissipativity of the system and identify a bounded absorbing set in the H1 phase space, which imposes intrinsic physical constraints and limits unphysical parameter exploration in large-scale optimization or black-box modeling. In addition, a Courant–Friedrichs–Lewy (CFL) stability condition is derived, providing a theoretical benchmark for time-step selection in numerical implementations, including physics-informed or hybrid neural architectures. This analytical framework supplies a mathematically controlled foundation for developing robust, interpretable, and stable pattern-recognition or time-series representations in complex geophysical systems.