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

How can neural operators be designed to better capture the long-time dynamics of dissipative time-dependent partial differential equations while remaining stable during long-horizon autoregressive rollouts? Standard neural operators like the Fourier Neural Operator (FNO) often lack physical interpretability and can suffer from training instability during multi-step forecasting because they treat dynamics in a black-box manner.

The authors propose the Inertial Manifold Neural Operator (IMNO), which integrates the principles of inertial manifold theory into operator learning. Rather than treating the entire function space uniformly, IMNO explicitly separates the PDE solution into a low-dimensional manifold component that captures dominant long-time dynamics and a residual component that accounts for out-of-manifold transient behavior. For shift-equivariant PDEs, the authors further develop IMNO-SE, a symmetry-preserving variant ensuring that spatial shifts in inputs induce corresponding spatial shifts in outputs.

By embedding the structural insights of reduced-order modeling and inertial manifolds into neural operators, IMNO achieves superior physical interpretability, higher accuracy, and robust stability in long-horizon autoregressive predictions. This bridges the gap between classical mathematical reduced-order models and modern data-driven scientific machine learning.