GA-NAR: Geometry-Adaptive Nonlinearity Allocation and Representation for Complex Physical Systems

Many complex physical systems exhibit a recurring dominant nonlinear-response structure whose local intensity and dominant/complement allocation vary across operating conditions, while standard supervised surrogates typically apply a fixed pattern of nonlinear computation. We introduce GA-NAR (Geometry-Adaptive Nonlinearity Allocation and Representation), a backbone-compatible framework that adapts neural nonlinear computation to the structure of the physical parameter-to-solution map. GA-NAR comprises two coupled components: Nonlinear Response Geometry Decomposition (NRGD), which identifies a recurring Dominant Nonlinear-Response Subspace together with state-dependent response intensity and subspace allocation, and Geometry-Adaptive Nonlinear Representation (GANR), which uses this information to organize nonlinear computation across subspaces and network depth. Independent experiments on NRGD across five systems verify reliable recovery of the proposed response descriptors, while separate no-physics and shared-physics factorial experiments show that GANR achieves the lowest NRMSE on four of five auxiliary systems under both conditions. The complete GA-NAR framework reduces standardized test MSE in all 18 activation–system comparisons, with median reductions of 11.5%, 2.1%, and 2.8% on IEEE-118, Kuramoto-64, and nonlinear heat FEM, respectively, alongside larger activation-specific gains in several cases. Physical residuals also decrease for most tested activations, showing that the predictive gains generally extend to physical consistency. We will release codes at https://anonymous.4open.science/r/GA-NAR.