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

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.

💡 Research Summary

The paper introduces DyMixOp, a novel neural operator designed to overcome two fundamental challenges in data‑driven PDE solving: (1) the difficulty of representing infinite‑dimensional, highly nonlinear dynamics in a tractable, finite‑dimensional form, and (2) the loss of high‑frequency information caused by the spectral bias of existing global‑only operators.
To address (1), the authors ground their method in inertial‑manifold theory and dynamics‑aware priors. They argue that many dissipative PDEs possess a low‑dimensional attracting manifold on which the essential linear structures and dominant nonlinear couplings reside. By projecting the original PDE flow onto this manifold, DyMixOp obtains a compact latent space that preserves the physics‑relevant interactions while dramatically reducing the number of degrees of freedom. This projection also provides a clear physical interpretation of the learned latent variables.
The second innovation is the Local‑Global Mixing (LGM) transformation. Traditional neural operators such as the Fourier Neural Operator (FNO) or Wavelet Neural Operator combine a global spectral branch with a local convolutional branch in an additive manner (LGA). While this mitigates spectral bias to some extent, the linear addition cannot reconstruct the multiplicative nonlinear couplings that dominate turbulent and chaotic flows. Inspired by the convective term u·∇u, LGM multiplies the output of a local fine‑scale feature extractor (large‑kernel convolution or continuous‑discrete convolution) with the output of a global spectral branch (Fourier or Laplace transform). This element‑wise product (Hadamard) creates a true nonlinear interaction between local and global information, preserving high‑frequency details and enabling the network to learn complex mode‑mixing dynamics.
Architecturally, DyMixOp stacks several LGM layers in a hybrid configuration. Each layer is equipped with a timescale‑adaptive gating mechanism that learns to weight the contribution of fast versus slow dynamics, and a parallel aggregation pathway that combines intermediate representations across layers. This design yields a flexible operator capable of handling a wide spectrum of PDE regimes—from diffusion‑dominated elliptic problems to convection‑dominated hyperbolic systems. Computationally, the global branch retains the O(N log N) complexity of FFT‑based methods, while the local branch adds only O(N) cost. Overall parameter count is reduced by roughly 30 % compared with standard FNO, leading to faster training and inference without sacrificing accuracy.
The authors evaluate DyMixOp on seven benchmark PDEs covering 1‑D to 3‑D domains, elliptic, parabolic, and hyperbolic types. Six of the seven tasks achieve state‑of‑the‑art performance, with DyMixOp outperforming DeepONet, FNO, GNO, and other recent operators. In chaotic regimes (e.g., Burgers’ equation with turbulent initial conditions), the method reduces the mean L2 error by up to 94.3 % relative to the best existing model. Moreover, DyMixOp demonstrates robustness to changes in mesh resolution and geometry, maintaining accuracy across uniform grids, unstructured point clouds, and multi‑resolution training setups. Training time on a single NVIDIA A100 GPU is approximately 1.2× faster than comparable FNO models, and inference scales linearly with the number of spatial points.
In summary, DyMixOp bridges complex dynamical systems theory and modern neural operator design. By embedding inertial‑manifold reduction and a multiplicative local‑global mixing mechanism, it delivers physically interpretable, high‑fidelity, and computationally efficient solutions for a broad class of PDEs. The work opens avenues for applying neural operators to high‑dimensional, chaotic, and multi‑scale scientific problems such as climate modeling, plasma dynamics, and advanced material simulations.