# Parametric Numerical Integration with (Differential) Machine Learning

arXiv: https://arxiv.org/abs/2512.11530 · HTML: https://mlsanity.com/paper/2512.11530/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 46 references that are in the map.
- First posted on arXiv 12 Dec 2025.

## Builds on (most cited first, 10 of 10)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Deep Hedging](https://mlsanity.com/paper/1802.03042/) (2018)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Probabilistic Numerics and Uncertainty in Computations](https://mlsanity.com/paper/1506.01326/) (2015)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [Differential Machine Learning](https://mlsanity.com/paper/2005.02347/) (2020)
- [Pricing options on flow forwards by neural networks in Hilbert space](https://mlsanity.com/paper/2202.11606/) (2022)
- [Deep Neural Network Solutions for Oscillatory Fredholm Integral Equations](https://mlsanity.com/paper/2401.07003/) (2024)

## Abstract (arXiv)

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.