Wolfram Computation Meets Knowledge

Wolfram Summer School


Tomasz Andrzejewski

Science and Technology

Class of 2019


Tomasz graduated from the University of Edinburgh in 2019 with a Master of Physics with Honors in Mathematical Physics. In the academic year 2017/18, he went on a student exchange to Caltech. As an undergraduate, he worked on kinematical Lie algebras, quantum information theory and machine learning. His current areas of interest include quantum gravity, quantum computing and machine learning. In the fall, he will be beginning his graduate program at the University of British Columbia, hoping to explore the intersection between quantum information theory and quantum gravity.

Computational Essay: Random Matrices and Quantum Chaos

Project: Improve Sampling for Monte Carlo Integration using Invertible Neural Networks


The goal of the project was to improve importance sampling used for Monte Carlo integration using an invertible neural network as a sampler. Invertible neural networks are state of the art when it comes to generating new samples of data. We focused on integrating sharp functions where it was easier to visualize the more important regions for sampling. In regions where the integrand is large, we should place a higher density of points.

Summary of Results

We implemented a function to improve sampling for Monte Carlo integration using an invertible neural network to sample. In particular, we explored a model called RealNVP and Invertible Residual Networks. With the latter, we spent some time trying to build in the neural network framework. We wrote a function to invert the residual network and a function to find a log-determinant of a matrix in order to get a loss function for training. Unfortunately, the log-determinant function did not pass the tests, so we switched to an already built-in model of invertible neural networks. Using RealNVP, we managed to improve sampling and tested it on different metrics, exploring the error of the integral as a function of the number of samples and their distribution.

Future Work

The future work could involve integrating different sharp functions and using different mixtures of distributions during sampling. We used a mixture of a uniform distribution and the one sampled with a neural network, but we have yet to explore a mixture of Gaussians. Another obvious development yet to be made is finishing the invertible residual network implementation. In the original paper, the loss used for training requires computing the log-determinant of a matrix, which proved to be rather challenging in the Wolfram Language neural net framework.