Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation
This provides a lightweight and practical alternative for solving PDEs in computational science, though it is incremental compared to existing neural operator methods.
The paper tackles learning solution operators for parametric PDEs by introducing the Monte Carlo-type Neural Operator (MCNO), which uses a Monte Carlo approach to approximate kernel integrals without spectral assumptions, achieving competitive accuracy on 1D benchmarks with low computational cost.
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.