Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916860885204992 |
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| author | Lee, Wonjae Kim, Taeyoung Park, Hyungbin |
| author_facet | Lee, Wonjae Kim, Taeyoung Park, Hyungbin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments Lee, Wonjae Kim, Taeyoung Park, Hyungbin Machine Learning Numerical Analysis 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. |
| title | Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2507.17887 |