ADAPTIVE MAMBA NEURAL OPERATORS

Traditional neural operators often struggle with irregular geometries or high-frequency features because their spectral bases lose orthogonality or fail to capture localized singularities. This paper asks whether a neural operator architecture can be designed to maintain the computational efficiency of state-space models (SSMs) while incorporating the mathematical rigor of adaptive spectral decomposition to better approximate complex PDE solutions.

The authors propose Adaptive Mamba Neural Operators (AMO), which replace the standard kernel integral formulation of SSMs with a reproducing kernel approach based on Takenaka-Malmquist (TM) systems. The architecture consists of three main components: a lifting operator that projects physical inputs into a latent space, a series of processing blocks that combine TM layers with bidirectional SSMs, and a projection operator that maps the latent representation back to the physical domain. The TM layers use a small MLP to predict 'poles' in the complex plane, which adaptively construct orthonormal bases. This design aligns the model with Adaptive Fourier Decomposition (AFD) theory, providing a principled way to handle irregular domains and long-range dependencies.

AMO consistently outperforms state-of-the-art neural operator solvers across a diverse set of benchmarks, including fluid dynamics (Navier-Stokes), solid physics (elasticity, plasticity), and finance (Black-Scholes). The model achieved an average relative L2 error reduction of 28.42% compared to existing baselines. The authors demonstrate that the use of adaptive poles allows the model to focus on rapidly changing regions of the solution manifold, while the TM-based construction ensures orthogonality, preventing the spectral mixing issues common in previous Mamba-based operators.