Parametric Numerical Integration with (Differential) Machine Learning

TL;DR

This paper presents a novel machine learning approach for parametric numerical integration, utilizing differential information to improve accuracy, scalability, and efficiency across various problem classes including statistical functionals, function approximation, and differential equations.

Contribution

Introduces a differential learning framework for parametric integrals that outperforms standard methods in accuracy and efficiency across multiple problem types.

Findings

Differential learning improves mean squared error over classical methods.

The approach enhances scalability and sample efficiency.

Effective across smooth and challenging numerical integrals.

Abstract

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.