🤖 AI Summary
This work addresses parametric numerical integration problems—including statistical functional evaluation, Chebyshev spectral approximation, and integrals arising from differential equations—by proposing a derivative-supervised differentiable machine learning framework. The method explicitly incorporates analytical derivative information into surrogate modeling of integrals for the first time, overcoming the limitations of conventional black-box regression while preserving physical consistency and substantially improving accuracy and generalization. Technically, it integrates deep neural networks with derivative-augmented training to enable efficient inference in high-dimensional parameter spaces. Evaluated on diverse benchmarks encompassing both smooth and ill-conditioned integrals, the approach achieves over 40% average reduction in mean squared error compared to standard architectures, while reducing sample requirements by a factor of 3–5. It thus delivers high accuracy, strong scalability, and exceptional sample efficiency.
📝 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.