Adaptive operator learning works on irregular 3D shapes and meshes
AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries
Computational Engineering, Finance, and Science
Summary
Scientists often need to solve complex math problems on oddly shaped 3D objects, but existing AI tools mostly work on simple grid-like shapes. The authors introduce AMOR, a new method that learns operators without needing regular grids and scales better to 3D cases. AMOR matches or beats existing methods on several problems and works well on realistic engineering shapes where others struggle. This makes it a useful tool for simulations involving complex geometries in science and engineering.
What this means in practice
- •For engineering simulation teams: Simulate physical systems on complex 3D shapes and irregular meshes efficiently without redesigning operator-learning models for each mesh type.
- •For computer graphics developers: Improve physics-based animations by applying learned operators on irregular meshes representing detailed 3D models.
Authors
David Voderholzer, Leonard Vetter, Thomas O'Leary-Roseberry
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 $m^3$ 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.