DyMixOp: A Neural Operator Designed from a Complex Dynamics Perspective with Local-Global Mixing for Solving PDEs

Abstract

A primary challenge in using neural networks to approximate nonlinear dynamical systems governed by partial differential equations (PDEs) lies in recasting these systems into a tractable representation particularly when the dynamics are inherently non-linearizable or require infinite-dimensional spaces for linearization. To address this challenge, we introduce DyMixOp, a novel neural operator framework for PDEs that integrates theoretical insights from complex dynamical systems. Grounded in dynamics-aware priors and inertial manifold theory, DyMixOp projects the original infinite-dimensional PDE dynamics onto a finite-dimensional latent space. This reduction preserves both essential linear structures and dominant nonlinear interactions, thereby establishing a physically interpretable and computationally structured foundation. Central to this approach is the local-global mixing (LGM) transformation, a key architectural innovation inspired by the convective nonlinearity in turbulent flows. By multiplicatively coupling local fine-scale features with global spectral information, LGM effectively captures high-frequency details and complex nonlinear couplings while mitigating the spectral bias that plagues many existing neural operators. The framework is further enhanced by a dynamics-informed architecture that stacks multiple LGM layers in a hybrid configuration, incorporating timescale-adaptive gating and parallel aggregation of intermediate dynamics. This design enables robust approximation of general evolutionary dynamics across diverse physical regimes. Extensive experiments on seven benchmark PDE systems spanning 1D to 3D, elliptic to hyperbolic types demonstrate that DyMixOp achieves state-of-the-art performance on six of them, significantly reducing prediction errors (by up to 94.3% in chaotic regimes) while maintaining computational efficiency and strong scalability.