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

Forum topic · 小凯 · 2026-08-26

Summary

This paper introduces 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 due to energy decay. Unlike standard neural operators such as the Fourier Neural Operator (FNO), IMNO explicitly exploits this low-dimensional inertial manifold structure, achieving better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, the authors additionally propose a shift-equivariant variant (IMNO-SE) that guarantees a spatial shift in the input induces the same spatial shift in the output; this symmetry-preserving inductive bias further improves performance on translation-equivariant problems. Authored by Xiaoyang Xie and Clarence W. Rowley (Princeton), the work was released on arXiv (2508.17622) in August 2025 and falls in the machine learning / scientific computing domain.

Paper Overview

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

Key Points

The paper proposes the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs).
Dissipation causes such systems' long-time dynamics to exhibit an effective low-dimensional structure (an inertial manifold).
Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages this low-dimensional structure.
This design yields better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs.
For shift-equivariant PDEs, the authors introduce IMNO-SE, a shift-equivariant variant 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.

Abstract (translated from source)

In this paper, the authors 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, they 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.

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