Structure-preserving neural networks for the regularized entropy-based closure of the Boltzmann moment system

The main challenge of large-scale numerical simulation of radiation transport is the high memory and computation time requirements of discretization methods for kinetic equations. In this work, we derive and investigate a neural network-based approximation to the entropy closure method to accurately compute the solution of the multi-dimensional moment system with a low memory footprint and competitive computational time. We extend methods developed for the standard entropy-based closure to the context of regularized entropy-based closures. The main idea is to interpret structure-preserving neural network approximations of the regularized entropy closure as a two-stage approximation to the original entropy closure. We conduct a numerical analysis of this approximation and investigate optimal parameter choices. Our numerical experiments demonstrate that the method has a much lower memory footprint than traditional methods with competitive computation times and simulation accuracy. The code and all trained networks are provided on GitHub https://github.com/ScSteffen/neuralEntropyClosures and https://github.com/CSMMLab/KiT-RT.

Rank-adaptive spectral pruning of convolutional layers during training

The computing cost and memory demand of deep learning pipelines have grown fast in recent years and thus a variety of pruning techniques have been developed to reduce model parameters. The majority of these techniques focus on reducing inference costs by pruning the network after a pass of full training. A smaller number of methods address the reduction of training costs, mostly based on compressing the network via low-rank layer factorizations. Despite their efficiency for linear layers, these methods fail to effectively handle convolutional filters. In this work, we propose a low-parametric training method that factorizes the convolutions into tensor Tucker format and adaptively prunes the Tucker ranks of the convolutional kernel during training. Leveraging fundamental results from geometric integration theory of differential equations on tensor manifolds, we obtain a robust training algorithm that provably approximates the full baseline performance and guarantees loss descent. A variety of experiments against the full model and alternative low-rank baselines are implemented, showing that the proposed method drastically reduces the training costs, while achieving high performance, comparable to or better than the full baseline, and consistently outperforms competing low-rank approaches.

Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them is significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. Moreover, our method automatically and dynamically adapts the ranks during training to achieve a desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.