Monte Carlo-Type Neural Operator for One-Dimensional Differential Equations

The Monte Carlo-type Neural Operator (MCNO) introduces a framework for learning solution operators of one-dimensional partial differential equations (PDEs) by directly learning the kernel function and approximating the associated integral operator using a Monte Carlo-type approach. Unlike Fourier Neural Operators (FNOs), which rely on spectral representations and assume translation-invariant kernels, MCNO makes no such assumptions.The kernel is represented as a learnable tensor over sampled input–output pairs, and sampling is performed once,uniformly at random from a discretized grid. This design enables generalization across multiple grid resolutions without relying on fixed global basis functions or repeated sampling during training, while an interpolation step maps between arbitrary input and output grids to further enhance flexibility. Experiments on standard 1D PDE benchmarks show that MCNO achieves competitive accuracy with efficient computational cost. More broadly, this work explores how Monte Carlo-type integration can be incorporated into neural operator frameworks for continuous-domain PDEs. It contributes to representation learning in function spaces and offers a simpler and practical alternative to spectral and graph-based approaches.