What question did this study set out to answer?

This research aims to introduce SEM-PDE, a new approach to surrogate modeling for parametric PDEs, focusing on interpretability and accuracy.

June 4, 2026Open Access

Structural Energy Model for PDE Surrogate Discovery

Key Points

This research aims to introduce SEM-PDE, a new approach to surrogate modeling for parametric PDEs, focusing on interpretability and accuracy.
Introduced SEM-PDE to discover mathematical formulas linking PDE parameters to solution structure.
Employed combinatorial search over physics-informed atom libraries and residual-driven atom genesis.
Evaluated SEM-PDE against eight families of PDEs including parabolic, hyperbolic, and nonlinear types.
Achieved publication-ready accuracy on six out of eight PDE families.
Built models 46 times faster than neural operator surrogates like FNO.
Provided robust extrapolation capabilities under distribution shift.

Abstract

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.

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