A Structure-Preserving Deep Galerkin Method for Nonlinear Coupled Reaction–Diffusion Systems

Accurate learning-based solvers for coupled reaction–diffusion systems should preserve mathematical structure rather than merely reduce pointwise residuals. We develop a structure-preserving deep Galerkin method that combines weak residual minimization with positivity, mass-balance, and energy-dissipation constraints. Adaptive quadrature concentrates test points near fronts and structural defects, while staged optimization stabilizes stiff training. Under coercivity, one-sided reaction monotonicity, and entropy compatibility, we establish an energy stability estimate, a weak-residual a posteriori error bound, and a unified structural reliability inequality. Numerical studies on pattern formation, nonlinear diffusion, and heterogeneous media show lower reported aggregate solution errors and structural drift than conventional deep Galerkin, variational, and physics-informed neural solvers; no inferential significance claim is made because seed-resolved records are not available in the retained archive. The results support invariant-aware variational learning as a reliable framework for nonlinear PDE approximation.