Latent Mamba Operator for Partial Differential Equations

TL;DR

The paper introduces the Latent Mamba Operator (LaMO), a novel neural operator that combines state-space models and kernel formulations to efficiently solve high-dimensional PDEs, achieving state-of-the-art results.

Contribution

LaMO is the first neural operator to integrate SSMs with kernel methods, improving scalability and accuracy in PDE solutions.

Findings

LaMO achieves 32.3% improvement over baselines in PDE solution approximation.

LaMO demonstrates consistent SOTA performance across diverse PDE benchmarks.

Theoretical connection established between SSMs and kernel integrals in neural operators.

Abstract

Neural operators have emerged as powerful data-driven frameworks for solving Partial Differential Equations (PDEs), offering significant speedups over numerical methods. However, existing neural operators struggle with scalability in high-dimensional spaces, incur high computational costs, and face challenges in capturing continuous and long-range dependencies in PDE dynamics. To address these limitations, we introduce the Latent Mamba Operator (LaMO), which integrates the efficiency of state-space models (SSMs) in latent space with the expressive power of kernel integral formulations in neural operators. We also establish a theoretical connection between state-space models (SSMs) and the kernel integral of neural operators. Extensive experiments across diverse PDE benchmarks on regular grids, structured meshes, and point clouds covering solid and fluid physics datasets, LaMOs achieve consistent state-of-the-art (SOTA) performance, with a 32.3% improvement over existing baselines in solution operator approximation, highlighting its efficacy in modeling complex PDE solutions.