用微分机器学习求解参数积分，效果优于传统方法。
Parametric Numerical Integration with (Differential) Machine Learning
- 引入包含导数信息的微分学习框架，提升训练效率。
- 在三类问题中均实现更低均方误差与更好样本效率。
- 适合需要高精度积分的科学计算与建模场景。
本文提出一种基于机器/深度学习的参数积分求解方法。除经典机器学习外，还采用融合导数信息的微分学习框架，突出其优势。研究涵盖三类典型问题：统计函数（包括矩和累积分布函数）、通过切比雪夫展开近似函数，以及直接来自微分方程的积分。这些例子从光滑闭式基准到复杂数值积分不等。在所有情况下，基于微分机器学习的方法均持续优于标准架构，实现更低均方误差、更强可扩展性与更高样本效率。
原文摘要 · Abstract (English)
In this work, we introduce a machine/deep learning methodology to solve parametric integrals. Besides classical machine learning approaches, we consider a differential learning framework that incorporates derivative information during training, emphasizing its advantageous properties. Our study covers three representative problem classes: statistical functionals (including moments and cumulative distribution functions), approximation of functions via Chebyshev expansions, and integrals arising directly from differential equations. These examples range from smooth closed-form benchmarks to challenging numerical integrals. Across all cases, the differential machine learning-based approach consistently outperforms standard architectures, achieving lower mean squared error, enhanced scalability, and improved sample efficiency.
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。