Structural Energy Model for PDE Surrogate Discovery

We introduce SEM-PDE, a surrogate modeling approach for parametric partial differential equations that discovers interpretable mathematical formulas connecting PDE parameters to solution structure. Unlike neural operator surrogates (FNO, DeepONet) that produce black-box predictions, SEM-PDE finds closed-form laws mapping parameters to SVD mode coefficients through combinatorial search over physics-informed atom libraries, augmented by residual-driven atom genesis. We evaluate on eight PDE families spanning parabolic, hyperbolic, elliptic, mixed, reaction-diffusion, frequency-domain, and dispersive nonlinear types. SEM-PDE achieves publication-ready accuracy on six of eight families, builds 46x faster than FNO, and provides robust extrapolation under distribution shift.