用蒙特卡洛方法直接近似微分方程解算子，无需频域假设。
Learning Solution Operators for Partial Differential Equations via Monte Carlo-Type Approximation
- 通过随机采样点的张量表示核函数，避免谱假设
- 在1D标准测试上达到与主流方法相当精度，计算开销低
- 支持多网格分辨率泛化，适合实际工程部署
蒙特卡洛型神经算子（MCNO）提出一种轻量级架构，通过蒙特卡洛方法直接近似参数化偏微分方程的核积分，无需频域或平移不变性假设。核函数以固定随机采样点上的可学习张量形式表示，该设计使模型能在不依赖固定全局基函数或训练时重复采样的情况下，实现对多种网格分辨率的泛化。在标准1D PDE基准测试中，MCNO表现出与谱型和图基神经算子相当的精度，且计算成本更低，为求解微分方程提供了一种简单实用的替代方案。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。