AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries
Abstract
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 m3 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 L2 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.
Related
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BibTeX
@misc{voderholzer2026amor,
title = {{AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries}},
author = {David Voderholzer and Leonard Vetter and Thomas O'Leary-Roseberry},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37485},
eprint = {2610.37485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37485},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries
AU - David Voderholzer
AU - Leonard Vetter
AU - Thomas O'Leary-Roseberry
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37485
UR - https://arxiv.org/abs/2610.37485
AB - 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.
ER -