Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation

TL;DR

The paper introduces MCNO, a lightweight neural operator that uses Monte Carlo sampling to learn solution operators for PDEs, offering flexibility and efficiency without spectral assumptions.

Contribution

MCNO is a novel neural operator architecture that approximates kernel integrals via Monte Carlo, enabling resolution-generalization without spectral constraints.

Findings

Achieves competitive accuracy on 1D PDE benchmarks

Offers low computational cost and simplicity

Does not rely on spectral or translation-invariance assumptions

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.