Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
2026-08-24 • Machine Learning
Machine Learning
AI summaryⓘ
The authors propose a new method called the Inertial Manifold Neural Operator (IMNO) to solve certain types of PDEs that change over time and tend to simplify into low-dimensional patterns. Unlike previous neural network methods, IMNO uses this low-dimensional structure to make more accurate and stable long-term predictions. They also created a version named IMNO-SE that respects spatial shifts, meaning moving the input results in a corresponding move in the output, which helps for specific PDEs with this property. The authors tested IMNO extensively and showed improved numerical performance.
Inertial ManifoldNeural OperatorPartial Differential EquationsDissipative SystemsLow-dimensional StructureFourier Neural OperatorShift-equivarianceAutoregressive TrainingLong-time Dynamics
Authors
Xiaoyang Xie, Clarence W. Rowley
Abstract
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 inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.