Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation
Year: 2025
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Choutri, Salah Eddine, Chauhan, Prajwal, Mazhar, Othmane, Jabari, Saif Eddin
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7106682127
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