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

TL;DR

MCNO introduces a Monte Carlo–type Neural Operator to learn PDE solution operators by approximating the kernel integral with a fixed set of random samples, avoiding Fourier transforms and deep hierarchical architectures. The model represents the kernel as a learnable per-sample tensor and combines feature mixing with a lightweight interpolation to produce full-grid outputs, yielding linear per-layer cost in the number of samples. Evaluated on 1D Burgers' and KdV benchmarks, MCNO achieves competitive accuracy with reduced computational cost and demonstrates robustness to grid resolution. The work positions MCNO as a practical alternative to spectral and graph-based neural operators and motivates future extensions to higher dimensions and adaptive sampling.

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.

Figures (5)

• Figure 1: MCNO architecture.
• Figure 2: Relative $L_2$ test loss vs Number of samples for Burgers
• Figure 3: Time per epoch vs Number of samples for Burgers
• Figure 4: Relative $L_2$ test loss vs Number of samples for KdV
• Figure 5: Time per epoch vs Number of samples for KdV