Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DyMixOp: Guiding Neural Operator Design for PDEs from a Complex Dynamics Perspective with Local-Global-Mixing

Pengyu Lai, Yixiao Chen, Hui Xu

TL;DR

This work addresses the challenge of solving nonlinear PDEs with neural operators by combining inertial-manifold-based dimensionality reduction with a convection-inspired Local-Global-Mixing transform. The resulting DyMixOp employs a dynamics-informed architecture to stack LGM layers that capture both linear and nonlinear latent dynamics in a finite-dimensional space, mitigating spectral bias and improving interpretability. Empirical results across KS, Darcy, NS, and SW benchmarks show state-of-the-art performance, especially in convection-dominated regimes, with favorable training efficiency and scalability. The approach offers a principled, physics-grounded path toward generalizable and efficient neural solvers for complex PDE dynamics.

Abstract

A primary challenge in using neural networks to approximate nonlinear dynamical systems governed by partial differential equations (PDEs) is transforming these systems into a suitable format, especially when dealing with non-linearizable dynamics or the need for infinite-dimensional spaces for linearization. This paper introduces DyMixOp, a novel neural operator framework for PDEs that integrates insights from complex dynamical systems to address this challenge. Grounded in inertial manifold theory, DyMixOp transforms infinite-dimensional nonlinear PDE dynamics into a finite-dimensional latent space, establishing a structured foundation that maintains essential nonlinear interactions and enhances physical interpretability. A key innovation is the Local-Global-Mixing (LGM) transformation, inspired by convection dynamics in turbulence. This transformation effectively captures both fine-scale details and nonlinear interactions, while mitigating spectral bias commonly found in existing neural operators. The framework is further strengthened by a dynamics-informed architecture that connects multiple LGM layers to approximate linear and nonlinear dynamics, reflecting the temporal evolution of dynamical systems. Experimental results across diverse PDE benchmarks demonstrate that DyMixOp achieves state-of-the-art performance, significantly reducing prediction errors, particularly in convection-dominated scenarios reaching up to 86.7\%, while maintaining computational efficiency and scalability.

DyMixOp: Guiding Neural Operator Design for PDEs from a Complex Dynamics Perspective with Local-Global-Mixing

TL;DR

This work addresses the challenge of solving nonlinear PDEs with neural operators by combining inertial-manifold-based dimensionality reduction with a convection-inspired Local-Global-Mixing transform. The resulting DyMixOp employs a dynamics-informed architecture to stack LGM layers that capture both linear and nonlinear latent dynamics in a finite-dimensional space, mitigating spectral bias and improving interpretability. Empirical results across KS, Darcy, NS, and SW benchmarks show state-of-the-art performance, especially in convection-dominated regimes, with favorable training efficiency and scalability. The approach offers a principled, physics-grounded path toward generalizable and efficient neural solvers for complex PDE dynamics.

Abstract

A primary challenge in using neural networks to approximate nonlinear dynamical systems governed by partial differential equations (PDEs) is transforming these systems into a suitable format, especially when dealing with non-linearizable dynamics or the need for infinite-dimensional spaces for linearization. This paper introduces DyMixOp, a novel neural operator framework for PDEs that integrates insights from complex dynamical systems to address this challenge. Grounded in inertial manifold theory, DyMixOp transforms infinite-dimensional nonlinear PDE dynamics into a finite-dimensional latent space, establishing a structured foundation that maintains essential nonlinear interactions and enhances physical interpretability. A key innovation is the Local-Global-Mixing (LGM) transformation, inspired by convection dynamics in turbulence. This transformation effectively captures both fine-scale details and nonlinear interactions, while mitigating spectral bias commonly found in existing neural operators. The framework is further strengthened by a dynamics-informed architecture that connects multiple LGM layers to approximate linear and nonlinear dynamics, reflecting the temporal evolution of dynamical systems. Experimental results across diverse PDE benchmarks demonstrate that DyMixOp achieves state-of-the-art performance, significantly reducing prediction errors, particularly in convection-dominated scenarios reaching up to 86.7\%, while maintaining computational efficiency and scalability.

Paper Structure

This paper contains 9 sections, 1 theorem, 16 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Challenges and Methodologies
  4. Results
  5. Experiment Settings
  6. Main Comparison Results
  7. Ablation and Scaling Experiments
  8. Conclusion
  9. Acknowledgments

Key Result

Proposition 1

(Principled Architectural Design) Our proposed neural operator, $\mathcal{G}^{\dagger}$, is constructed as a direct functional analogue of the reduced dynamics dictated by inertial manifold theory. By projecting Equation dynamics_only_associated_with_reduced_state onto a spectral basis to obtain dyn Specifically, the architecture $\mathcal{G}^{\dagger}$ is structured to explicitly model the compon

Figures (3)

  • Figure 1: Illustration of the DyMixOp model composed of the dimension-shifting layer and its inverse, the projection layer and its inverse, and the LGM layers that adopt LGM transformations to approximate dynamics within the dynamics-informed architecture.
  • Figure 2: The performance of all model configurations across all datasets is visualized in a plane defined by training time per epoch and GPU memory usage. The size of each point indicates model performance, quantified as 1000 divided by the test dataset loss.
  • Figure 3: The illustration of scaling experiments for training samples and model sizes and ablation experiments. 'W' is the channel width and 'L' is the number of layers.

Theorems & Definitions (3)

  • Proposition 1
  • Definition 1
  • Definition 2