用低维结构提升非线性耗散PDE的长期预测精度与稳定性
Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
- 基于惯性流形设计神经算子，显式利用系统低维动态结构
- 在长时序预测中误差显著低于FNO，且训练更稳定
- 适合需要物理可解释性的耗散型动力系统建模
本文提出惯性流形神经算子（IMNO），用于求解耗散型时变偏微分方程。由于耗散作用，这类系统的长时间动力学通常呈现有效的低维结构。与标准神经算子如傅里叶神经算子（FNO）不同，IMNO 显式利用该低维结构，在非线性耗散PDE的长期自回归训练与预测中，实现了更高的物理可解释性、准确性和稳定性。针对平移等变的PDE，进一步提出平移等变变体（IMNO-SE），保证输入的空间平移对应输出的空间平移，通过保持对称性的归纳偏置，显著提升了在平移等变PDE上的性能。通过大量基准实验验证了IMNO的数值表现。
原文摘要 · Abstract (English)
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.
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