AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries
David Voderholzer, Leonard Vetter, Thomas O'Leary-Roseberry
Parametric PDEs in science and engineering are often posed on irregular meshes with geometries that may contain holes or complex boundaries, yet popular operator-surrogate architectures such as the Fourier Neural Operator (FNO) rely on structured geometries for fast computational kernels. We introduce Adaptive Modal Operator Refinement (AMOR), which replaces FNO's Fourier-based spectral convolutions with reduced-basis blocks and supports both data-driven and prescribed input/output bases through a common encode/decode interface. AMOR thus applies unmodified to arbitrary meshes and, because its spectral parameterization has no tensor-product dependence on the spatial dimension, it scales favorably to three dimensions, where FNO's spectral weights grow as $m^3$ with the number of spectral modes $m$. We prove that AMOR is a universal approximator of continuous operators. On two-dimensional structured-grid benchmarks, AMOR is competitive with FNO overall and surpasses PCANet: it outperforms FNO on nonlinear diffusion and Helmholtz problems, while a static snapshot basis keeps it from matching FNO's accuracy on the time-dependent Kolmogorov flow problem. On three-dimensional problems it matches or exceeds FNO's accuracy at lower inference cost on linear elastostatics while trailing FNO on a turbulent lid-driven-cavity flow and scales favorably to problems demanding high capacity. On irregular meshes, where FNO is not directly applicable, AMOR reduces the relative $L^2$ error over the FNO extension Geo-FNO by $31\%$ (airfoil) and $42\%$ (elasticity). Finally, AMOR maintains strong performance across three physically distinct, experimentally motivated loading regimes on a topologically nontrivial three-dimensional aircraft-rib mesh, demonstrating its applicability to realistic engineering problems without modification.
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