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

2025-11-24

AI Summary

This work addresses key limitations in learning solution operators for parametric partial differential equations (PDEs)—namely, reliance on spectral assumptions, translation invariance, and fixed discretization grids. To overcome these, we propose a lightweight neural operator architecture that abandons Fourier transforms and predefined basis expansions. Instead, it employs learnable tensors to directly parameterize the integral kernel and approximates the kernel integral via Monte Carlo sampling over arbitrarily distributed point clouds, enabling resolution-agnostic generalization. The design eliminates repeated grid sampling, yielding a compact, computationally efficient, and easily scalable framework. On standard 1D PDE benchmarks, our method achieves accuracy comparable to state-of-the-art approaches while significantly reducing inference cost. Experimental results demonstrate its superior balance of accuracy, computational efficiency, and cross-resolution generalizability.

Technology Category

Machine Learning: Learning with ManifoldsSearch and Optimization: Learning to SearchComputer Vision: Learning & Optimization for CV

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsUser Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendationSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Learning solution operators for parametric PDEs via Monte Carlo approximation
• Developing grid-independent neural operators without spectral assumptions
• Achieving computational efficiency while maintaining competitive accuracy

Innovation

• Monte Carlo approach approximates kernel integral directly
• Learnable tensor kernel without spectral assumptions
• Generalizes across grid resolutions without fixed basis