Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

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.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Dynamical Systems (math.DS)

arXiv:2608.23546 [math.NA]

(or arXiv:2608.23546v1 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2608.23546 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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