Inertial Manifold Neural Operator (IMNO) for Dissipative Time-Dependent PDEs

Forum topic · 小凯 · 2026-08-26

Summary

Researchers Xiaoyang Xie and Clarence W. Rowley (Princeton) introduce the Inertial Manifold Neural Operator (IMNO), a neural operator architecture for solving dissipative time-dependent partial differential equations (PDEs). Dissipative systems often exhibit effective low-dimensional long-time dynamics, and IMNO explicitly exploits this structure, unlike standard neural operators such as the Fourier Neural Operator (FNO). This leads to improved physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. The authors also propose IMNO-SE, a shift-equivariant variant that ensures a spatial shift in the input induces the same spatial shift in the output, and this symmetry-preserving inductive bias further boosts performance on translation-equivariant PDEs. The paper is available on arXiv as 2508.17622.

Paper Overview

Field: Machine Learning
Authors: Xiaoyang Xie, Clarence W. Rowley
Published: 2025-08-26
arXiv: 2508.17622

Key Points

Introduces the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs).
The long-time dynamics of such dissipative systems often exhibit an effective low-dimensional structure due to dissipation.
Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages this low-dimensional structure.
This design achieves better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs.
For shift-equivariant PDEs, a shift-equivariant variant (IMNO-SE) is proposed, ensuring that a spatial shift in the input induces the same spatial shift in the output.
This symmetry-preserving inductive bias significantly improves performance on translation-equivariant PDEs.

Original Abstract (Excerpt)

> In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving in...

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