Adaptive Mamba Neural Operators
2026-07-20 • Machine Learning
Machine Learning
AI summaryⓘ
The authors introduce Adaptive Mamba Neural Operators (AMO), a new method to solve partial differential equations (PDEs) on different shapes and types of meshes. Instead of traditional kernel integral methods, they use reproducing kernels and special mathematical systems called Takenaka-Malmquist systems to better capture solutions. Their approach aligns with adaptive Fourier decomposition theory, allowing it to work well on complex geometries. In tests across physics and finance problems, AMO showed more accurate results compared to current best methods. This suggests a new, more understandable way to build neural networks for solving PDEs.
Partial Differential EquationsNeural OperatorsReproducing KernelsState-Space ModelsTakenaka-Malmquist SystemsAdaptive Fourier DecompositionSolution ManifoldMeshesBenchmark ProblemsRelative L2 Error
Authors
Zeyuan Song, Zheyu Jiang
Abstract
Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.