Geomechanics-motivated discovery of three-dimensional coupled governing equations via physics-informed symbolic genetic algorithms

Geomechanical processes commonly involve strong multiphysics coupling and pronounced scale disparity, while the governing equations used in practice are often prescribed a priori and calibrated from limited observations. When the assumed analytical form is misspecified, such template-based modeling can introduce structural bias. This study addresses the discovery of explicit three-dimensional coupled partial differential equations from sparse observations and proposes a Physics-Informed Symbolic Genetic Algorithm (PI-SGA) to improve tractability, identifiability, and physical admissibility in strongly coupled systems. The framework combines three components: (i) a tensor-consistent operator library that restricts symbolic search to frame-consistent candidates; (ii) a field-separation strategy that reformulates coupled discovery into single-field regressions with measured exogenous coupling inputs; and (iii) a stochastic spatiotemporal collocation scheme that reduces the cost of three-dimensional discovery while preserving the dynamics required for identification. PI-SGA is evaluated on three benchmark systems with distinct coupled behaviors: Biot poroelasticity, incompressible Navier–Stokes flow, and a Turing reaction–diffusion system. Across these cases, the framework recovers compact governing structures with coefficients consistent with the reference models under sparse sampling. Sensitivity analyses under 0–10% observational noise show stable structural recovery with benchmark-dependent coefficient sensitivity, and comparison with PySINDy indicates improved robustness in coupled and noise-sensitive settings, albeit at increased computational cost. Additional forward validation under shifted initial conditions, while preserving the original boundary-condition class, further supports the physical fidelity of the recovered equations within the tested benchmark regime. The proposed workflow provides a physically constrained and interpretable baseline for discovering three-dimensional coupled governing equations in geomechanics-motivated applications.