Parametric Numerical Integration with (Differential) Machine Learning
This work addresses computational challenges in numerical integration for applications like statistical analysis and differential equations, representing a novel method for a known bottleneck rather than a paradigm shift.
The authors tackled the problem of solving parametric integrals by introducing a differential machine learning framework that incorporates derivative information during training. Their approach consistently outperformed standard architectures across three problem classes, achieving lower mean squared error, enhanced scalability, and improved sample efficiency.
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.