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.

翻译：本文提出了一种利用机器学习/深度学习技术求解参数化积分的方法。除了经典的机器学习方法外，我们引入了一种微分学习框架，该框架在训练过程中融合了导数信息，并重点阐述了其优势特性。本研究涵盖了三类代表性课题：统计泛函（包括矩量与累积分布函数）、基于切比雪夫展开的函数逼近，以及微分方程直接导出的积分问题。这些算例涵盖了从光滑闭式基准测试到具有挑战性的数值积分等多种情形。在所有案例中，基于微分机器学习的方法均持续优于标准架构，实现了更低的均方误差、更强的可扩展性以及更高的样本效率。