Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911283568181248 |
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| author | Choutri, Salah Eddine Chauhan, Prajwal Mazhar, Othmane Jabari, Saif Eddin |
| author_facet | Choutri, Salah Eddine Chauhan, Prajwal Mazhar, Othmane Jabari, Saif Eddin |
| contents | The Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural Operators, MCNO makes no spectral or translation-invariance assumptions. The kernel is represented as a learnable tensor over a fixed set of randomly sampled points. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with low computational cost, providing a simple and practical alternative to spectral and graph-based neural operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation Choutri, Salah Eddine Chauhan, Prajwal Mazhar, Othmane Jabari, Saif Eddin Machine Learning Artificial Intelligence The Monte Carlo-type Neural Operator (MCNO) introduces a lightweight architecture for learning solution operators for parametric PDEs by directly approximating the kernel integral using a Monte Carlo approach. Unlike Fourier Neural Operators, MCNO makes no spectral or translation-invariance assumptions. The kernel is represented as a learnable tensor over a fixed set of randomly sampled points. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with low computational cost, providing a simple and practical alternative to spectral and graph-based neural operators. |
| title | Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2511.18930 |