A machine learning approach for efficient multi-dimensional integration.

Area of Science:

• Computational Physics
• Numerical Analysis
• Machine Learning Applications

Background:

• Many physics problems require multi-dimensional integration where analytical solutions are unavailable.
• Traditional numerical integration methods can be computationally expensive, necessitating efficient algorithms.

Purpose of the Study:

• To develop a novel, efficient machine learning (ML) algorithm for numerical multi-dimensional integration in physics.
• To improve the accuracy and reduce the computational cost of evaluating complex integrals.

Main Methods:

• Training ML regression models (multi-layer perceptron, gradient boosting, Gaussian process) to approximate integrands.
• Implementing a bias correction mechanism using the difference between ML approximation and true integral values.
• Investigating performance across six diverse integrand families and varying dimensions.

Main Results:

• The proposed ML algorithm achieves integral estimates with over an order of magnitude smaller uncertainties than the VEGAS algorithm for the same number of evaluations.
• Demonstrated effectiveness across various dimensions and integrand complexities typical in physics problems.
• The bias correction ensures unbiased estimates with statistically sound error estimations.

Conclusions:

• The novel ML-based numerical integration algorithm offers a significant improvement in efficiency and accuracy over existing methods.
• This approach provides a powerful tool for tackling computationally intensive integration problems in physics.
• The bias correction strategy is crucial for achieving unbiased results and reliable error quantification.