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Quantifying uncertainty in weather forecasts is critical, especially for predicting extreme weather events. This is typically accomplished with ensemble prediction systems, which consist of many perturbed numerical weather simulations, or trajectories, run in parallel. These systems are associated with a high computational cost and often involve statistical post-processing steps to inexpensively improve their raw prediction qualities. We propose a mixed model that uses only a subset of the original weather trajectories combined with a post-processing step using deep neural networks. These enable the model to account for non-linear relationships that are not captured by current numerical models or post-processing methods. Applied to the global data, our mixed models achieve a relative improvement in ensemble forecast skill (CRPS) of over 14%. Furthermore, we demonstrate that the improvement is larger for extreme weather events on select case studies. We also show that our post-processing can use fewer trajectories to achieve comparable results to the full ensemble. By using fewer trajectories, the computational costs of an ensemble prediction system can be reduced, allowing it to run at higher resolution and produce more accurate forecasts. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.

The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.. The supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience -Albert Einstein, 1933 INTRODUCTIONDeep learning shows unparalleled promise for solving very complex real-world problems in areas such as computer vision, natural language processing, knowledge representation, recommendation systems, drug discovery, and many more. With this development, the field of machine learning is moving from traditional feature engineering to neural architecture engineering. However, still little is known about how to pick the right architecture to solve a specific task. Several methods such as translational equivariance in convolutional layers, recurrence, structured weight sharing, pooling, or locality are used to introduce strong inductive biases in the model design. Yet, the exact model size and capacity required for a task remain unknown and a common strategy is to train overparameterized models and compress them into smaller representations. Biological brains, especially the human brain, are hierarchical, sparse, and recurrent structures [Friston 2008] and one can draw some similarities with the inductive biases in today's artificial neural networks. Sparsity plays an important role in scaling biological brains-the more

This paper introduces a new approach to building distributedmemory graph analytics systems that exploits heterogeneity in processor types (CPU and GPU), partitioning policies, and programming models. The key to this approach is Gluon, a communication-optimizing substrate. Programmers write applications in a shared-memory programming system of their choice and interface these applications with Gluon using a lightweight API. Gluon enables these programs to run on heterogeneous clusters and optimizes communication in a novel way by exploiting structural and temporal invariants of graph partitioning policies. To demonstrate Gluon's ability to support different programming models, we interfaced Gluon with the Galois and Ligra shared-memory graph analytics systems to produce distributed-memory versions of these systems named D-Galois and D-Ligra, respectively. To demonstrate Gluon's ability to support heterogeneous processors, we interfaced Gluon with IrGL, a state-of-the-art single-GPU system for * Both authors contributed equally.

This paper introduces a new approach to building distributedmemory graph analytics systems that exploits heterogeneity in processor types (CPU and GPU), partitioning policies, and programming models. The key to this approach is Gluon, a communication-optimizing substrate. Programmers write applications in a shared-memory programming system of their choice and interface these applications with Gluon using a lightweight API. Gluon enables these programs to run on heterogeneous clusters and optimizes communication in a novel way by exploiting structural and temporal invariants of graph partitioning policies. To demonstrate Gluon's ability to support different programming models, we interfaced Gluon with the Galois and Ligra shared-memory graph analytics systems to produce distributed-memory versions of these systems named D-Galois and D-Ligra, respectively. To demonstrate Gluon's ability to support heterogeneous processors, we interfaced Gluon with IrGL, a state-of-the-art single-GPU system for * Both authors contributed equally.

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B. Performance modelingWe make use of analytic models for some of our performance estimates, particularly communication. We use a linear model [21] for communication, where α is the latency and β is the inverse bandwidth. Then the cost to send a message between two nodes is α + βn. We additionally assume that the network is full-duplex and that there is no interference.Collective communication operations such as allreduce will be important for some operations; for these, we use the performance models of Thakur et al [22]. For distributed matrix multiplication, we use the performance models developed for the Elemental library [23]. C. NotationWe now define some notation for distributed tensors that will be used throughout this paper. Our notation is heavily based on the tensor notation developed for the FLAME project [23]- [25].A tensor is an M -dimensional array, where the size of dimension m is I m , and we write I = (I 0 , . . . , I M −1 ) to refer to the shape of an entire tensor.

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