MFNO模型提升非马尔可夫过程建模能力，兼具理论保障与高效生成。
Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments
- 引入镜像填充的傅里叶神经算子，处理非周期性输入
- 理论上可任意精度逼近路径依赖型随机微分方程解
- 生成速度远超传统数值方法，适合高维时序建模
本文提出一种基于算子的神经网络——镜像填充傅里叶神经算子（MFNO），用于学习随机系统的动态行为。MFNO通过引入镜像填充机制，扩展了标准傅里叶神经算子（FNO）对非周期性输入的处理能力。我们严格证明，MFNO可任意精度逼近路径依赖型随机微分方程的解以及分数布朗运动的Lipschitz变换。理论分析基于Wong--Zakai型定理及多种逼近技术。实验表明，MFNO展现出强大的分辨率泛化能力，这是标准架构如LSTM、TCN和DeepONet罕见的特性。此外，该模型在性能上可媲美或优于基线方法，同时样本路径生成速度显著快于经典数值方案。
原文摘要 · Abstract (English)
This paper introduces an operator-based neural network, the mirror-padded Fourier neural operator (MFNO), designed to learn the dynamics of stochastic systems. MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror padding, enabling it to handle non-periodic inputs. We rigorously prove that MFNOs can approximate solutions of path-dependent stochastic differential equations and Lipschitz transformations of fractional Brownian motions to an arbitrary degree of accuracy. Our theoretical analysis builds on Wong--Zakai type theorems and various approximation techniques. Empirically, the MFNO exhibits strong resolution generalization--a property rarely seen in standard architectures such as LSTMs, TCNs, and DeepONet. Furthermore, our model achieves performance that is comparable or superior to these baselines while offering significantly faster sample path generation than classical numerical schemes.
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