Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911498939400192 |
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| author | Shi, Dai Thompson, Luke Han, Andi Hu, Peiyan Gao, Junbin Hernández-Lobato, José Miguel |
| author_facet | Shi, Dai Thompson, Luke Han, Andi Hu, Peiyan Gao, Junbin Hernández-Lobato, José Miguel |
| contents | In this paper, we explore how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular stochastic partial differential equations, e.g., the dynamic $\boldsymbolΦ^4_2$ model simulated in the recent works. Unlike the previous WCE-NO which solves SPDEs by simply inserting Wick-Hermite features into the backbone NO model, we leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder. The resulting WCE-FiLM-NO shows excellent performance on $\boldsymbolΦ^4_2$, as measured by relative $L_2$ loss, out-of-distribution $L_2$ loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating $\boldsymbolΦ^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory. To the best of our knowledge, this is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbolΦ^4_3$ model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_08219 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations Shi, Dai Thompson, Luke Han, Andi Hu, Peiyan Gao, Junbin Hernández-Lobato, José Miguel Machine Learning In this paper, we explore how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular stochastic partial differential equations, e.g., the dynamic $\boldsymbolΦ^4_2$ model simulated in the recent works. Unlike the previous WCE-NO which solves SPDEs by simply inserting Wick-Hermite features into the backbone NO model, we leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder. The resulting WCE-FiLM-NO shows excellent performance on $\boldsymbolΦ^4_2$, as measured by relative $L_2$ loss, out-of-distribution $L_2$ loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating $\boldsymbolΦ^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory. To the best of our knowledge, this is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbolΦ^4_3$ model. |
| title | Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.08219 |