用于耗散型时变偏微分方程的惯性流形神经算子
Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
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中文总结 AI 辅助
该研究提出惯性流形神经算子（IMNO）及其平移等变变体IMNO-SE，利用耗散型时变PDE的低维结构提升长时预测性能，通过基准实验验证了其优势。
中文摘要 AI 辅助
本文提出用于求解耗散型时变偏微分方程（PDE）的惯性流形神经算子（IMNO）。这类系统的长期动力学因耗散往往呈现出有效的低维结构。与傅里叶神经算子（FNO）等标准神经算子架构不同，IMNO明确利用该低维结构，在非线性耗散PDE的长自回归训练与预测中实现更优的物理解释性、精度和稳定性。对于平移等变PDE，我们进一步提出所提神经算子的平移等变变体（IMNO-SE），确保输入的空间平移会在输出中诱导相同的空间平移。这种保持对称性的归纳偏置大幅提升了其在平移等变PDE中的性能。本文开展了大量基准实验以数值评估IMNO的性能。
英文摘要
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.
发表机构
- Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。
