这篇论文提出了IMNO，一个解决耗散PDEs的新方法，比传统神经网络操作符更稳定准确，特别适合对耗散系统的研究。
This paper introduces the Inertial Manifold Neural Operator (IMNO) for solving dissipative PDEs, offering better interpretability, accuracy, and stability. IMNO-SE, a variant for shift-equivariant PDEs, preserves symmetry and improves performance. Benchmark experiments demonstrate IMNO's effectiveness.
Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
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.