Neural Operator-Accelerated High-Dimensional Integration
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| Natura: | Recurso digital |
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2025
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | The numerical evaluation of high-dimensional integrals is a fundamental challenge across science and engineering, frequently bottlenecked by the curse of dimensionality. Traditional quadrature methods exhibit computational costs that scale exponentially with the dimension of the integrand, rendering many important problems intractable. This paper introduces a novel framework for accelerating high-dimensional integration by leveraging the expressive power of Neural Operators, specifically the Fourier Neural Operator (FNO). Unlike conventional neural networks that learn mappings between finite-dimensional spaces, Neural Operators learn mappings between infinite-dimensional function spaces. We frame the problem of numerical integration as learning an operator that maps an integrand function to its definite integral. By training an FNO on a dataset of function-integral pairs, the model learns a generalized representation of the integration operator itself. This approach is discretization-invariant, allowing the trained operator to evaluate integrals of new functions sampled at various resolutions without retraining. We demonstrate that this method can effectively mitigate the curse of dimensionality for certain classes of smooth functions. The methodology involves representing integrand functions on a uniform grid, processing them through a series of Fourier layers that perform global convolutions efficiently in the frequency domain, and mapping the result to a single scalar value representing the integral. We present theoretical arguments for the operator's approximation capabilities and provide numerical results on synthetic high-dimensional benchmark problems, showing significant computational speed-up and comparable accuracy to Monte Carlo methods, particularly for functions with underlying structural regularities. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17444807 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Neural Operator-Accelerated High-Dimensional Integration SÉRGIO DE ANDRADE, PAULO Neural Operator High-Dimensional Integration Fourier Neural Operator Curse of Dimensionality Scientific Machine Learning Numerical Quadrature The numerical evaluation of high-dimensional integrals is a fundamental challenge across science and engineering, frequently bottlenecked by the curse of dimensionality. Traditional quadrature methods exhibit computational costs that scale exponentially with the dimension of the integrand, rendering many important problems intractable. This paper introduces a novel framework for accelerating high-dimensional integration by leveraging the expressive power of Neural Operators, specifically the Fourier Neural Operator (FNO). Unlike conventional neural networks that learn mappings between finite-dimensional spaces, Neural Operators learn mappings between infinite-dimensional function spaces. We frame the problem of numerical integration as learning an operator that maps an integrand function to its definite integral. By training an FNO on a dataset of function-integral pairs, the model learns a generalized representation of the integration operator itself. This approach is discretization-invariant, allowing the trained operator to evaluate integrals of new functions sampled at various resolutions without retraining. We demonstrate that this method can effectively mitigate the curse of dimensionality for certain classes of smooth functions. The methodology involves representing integrand functions on a uniform grid, processing them through a series of Fourier layers that perform global convolutions efficiently in the frequency domain, and mapping the result to a single scalar value representing the integral. We present theoretical arguments for the operator's approximation capabilities and provide numerical results on synthetic high-dimensional benchmark problems, showing significant computational speed-up and comparable accuracy to Monte Carlo methods, particularly for functions with underlying structural regularities. |
| title | Neural Operator-Accelerated High-Dimensional Integration |
| topic | Neural Operator High-Dimensional Integration Fourier Neural Operator Curse of Dimensionality Scientific Machine Learning Numerical Quadrature |
| url | https://doi.org/10.5281/zenodo.17444807 |