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Frequently in socio-environmental sciences, models are used as tools to represent, understand, project and predict the behaviour of these complex systems. Along the modelling chain, Good Modelling Practices have been evolving that ensure -- amongst others -- that models are transparent and their results replicable. Whenever such models are represented in software, Good Modelling meet Good Software Practices, such as a tractable development workflow, good code, collaborative development and governance, continuous integration and deployment; and they meet Good Scientific Practices, such as attribution of copyrights and acknowledgement of intellectual property, publication of a software paper and archiving. Too often in existing socio-environmental model software, these practices have been regarded as an add-on to be considered at a later stage only; modellers have shied away from publishing their model as open source out of fear that having to add good practices is too demanding. We here argue for making a habit of following a list of simple and not so simple practices early on in the implementation of the model life cycle. We contextualise cherry-picked and hands-on practices for supporting Good Modelling Practice, and we demonstrate their application in the example context of the Viable North Sea fisheries socio-ecological systems model.

Existing symbolic regression methods organize the space of candidate mathematical expressions primarily based on their syntactic, structural similarity. However, this approach overlooks crucial equivalences between expressions that arise from mathematical symmetries, such as commutativity, associativity, and distribution laws for arithmetic operations. Consequently, expressions with similar errors on a given data set are apart from each other in the search space. This leads to a rough error landscape in the search space that efficient local, gradient-based methods cannot explore. This paper proposes and implements a measure of a behavioral distance, BED, that clusters together expressions with similar errors. The experimental results show that the stochastic method for calculating BED achieves consistency with a modest number of sampled values for evaluating the expressions. This leads to computational efficiency comparable to the tree-based syntactic distance. Our findings also reveal that BED significantly improves the smoothness of the error landscape in the search space for symbolic regression.

Hamiltonian neural networks (HNNs) are state-of-the-art models that regress the vector field of a dynamical system under the learning bias of Hamilton's equations. A recent observation is that embedding a bias regarding the additive separability of the Hamiltonian reduces the regression complexity and improves regression performance. We propose separable HNNs that embed additive separability within HNNs using observational, learning, and inductive biases. We show that the proposed models are more effective than the HNN at regressing the Hamiltonian and the vector field. Consequently, the proposed models predict the dynamics and conserve the total energy of the Hamiltonian system more accurately.

Connected decision boundaries are useful in several tasks like image segmentation, clustering, alpha-shape or defining a region in nD-space. However, the machine learning literature lacks methods for generating connected decision boundaries using neural networks. Thresholding an invex function, a generalization of a convex function, generates such decision boundaries. This paper presents two methods for constructing invex functions using neural networks. The first approach is based on constraining a neural network with Gradient Clipped-Gradient Penality (GCGP), where we clip and penalise the gradients. In contrast, the second one is based on the relationship of the invex function to the composition of invertible and convex functions. We employ connectedness as a basic interpretation method and create connected region-based classifiers. We show that multiple connected set based classifiers can approximate any classification function. In the experiments section, we use our methods for classification tasks using an ensemble of 1-vs-all models as well as using a single multiclass model on small-scale datasets. The experiments show that connected set-based classifiers do not pose any disadvantage over ordinary neural network classifiers, but rather, enhance their interpretability. We also did an extensive study on the properties of invex function and connected sets for interpretability and network morphism with experiments on toy and real-world data sets. Our study suggests that invex function is fundamental to understanding and applying locality and connectedness of input space which is useful for various downstream tasks.

This paper presents a new approach for assembling graph neural networks based on framelet transforms. The latter provides a multi-scale representation for graph-structured data. With the framelet system, we can decompose the graph feature into low-pass and high-pass frequencies as extracted features for network training, which then defines a framelet-based graph convolution. The framelet decomposition naturally induces a graph pooling strategy by aggregating the graph feature into low-pass and high-pass spectra, which considers both the feature values and geometry of the graph data and conserves the total information. The graph neural networks with the proposed framelet convolution and pooling achieve state-of-the-art performance in many types of node and graph prediction tasks. Moreover, we propose shrinkage as a new activation for the framelet convolution, which thresholds the high-frequency information at different scales. Compared to ReLU, shrinkage in framelet convolution improves the graph neural network model in terms of denoising and signal compression: noises in both node and structure can be significantly reduced by accurately cutting off the high-pass coefficients from framelet decomposition, and the signal can be compressed to less than half its original size with the prediction performance well preserved.

The aim of this work is to develop a fully-distributed algorithmic framework for training graph convolutional networks (GCNs). The proposed method is able to exploit the meaningful relational structure of the input data, which are collected by a set of agents that communicate over a sparse network topology. After formulating the centralized GCN training problem, we first show how to make inference in a distributed scenario where the underlying data graph is split among different agents. Then, we propose a distributed gradient descent procedure to solve the GCN training problem. The resulting model distributes computation along three lines: during inference, during back-propagation, and during optimization. Convergence to stationary solutions of the GCN training problem is also established under mild conditions. Finally, we propose an optimization criterion to design the communication topology between agents in order to match with the graph describing data relationships. A wide set of numerical results validate our proposal. To the best of our knowledge, this is the first work combining graph convolutional neural networks with distributed optimization.

Knowledge graph (KG) embedding encodes the entities and relations from a KG into low-dimensional vector spaces to support various applications such as KG completion, question answering, and recommender systems. In real world, knowledge graphs (KGs) are dynamic and evolve over time with addition or deletion of triples. However, most existing models focus on embedding static KGs while neglecting dynamics. To adapt to the changes in a KG, these models need to be re-trained on the whole KG with a high time cost. In this paper, to tackle the aforementioned problem, we propose a new context-aware Dynamic Knowledge Graph Embedding (DKGE) method which supports the embedding learning in an online fashion. DKGE introduces two different representations (i.e., knowledge embedding and contextual element embedding) for each entity and each relation, in the joint modeling of entities and relations as well as their contexts, by employing two attentive graph convolutional networks, a gate strategy, and translation operations. This effectively helps limit the impacts of a KG update in certain regions, not in the entire graph, so that DKGE can rapidly acquire the updated KG embedding by a proposed online learning algorithm. Furthermore, DKGE can also learn KG embedding from scratch. Experiments on the tasks of link prediction and question answering in a dynamic environment demonstrate the effectiveness and efficiency of DKGE.

Embedding models for deterministic Knowledge Graphs (KG) have been extensively studied, with the purpose of capturing latent semantic relations between entities and incorporating the structured knowledge into machine learning. However, there are many KGs that model uncertain knowledge, which typically model the inherent uncertainty of relations facts with a confidence score, and embedding such uncertain knowledge represents an unresolved challenge. The capturing of uncertain knowledge will benefit many knowledge-driven applications such as question answering and semantic search by providing more natural characterization of the knowledge. In this paper, we propose a novel uncertain KG embedding model UKGE, which aims to preserve both structural and uncertainty information of relation facts in the embedding space. Unlike previous models that characterize relation facts with binary classification techniques, UKGE learns embeddings according to the confidence scores of uncertain relation facts. To further enhance the precision of UKGE, we also introduce probabilistic soft logic to infer confidence scores for unseen relation facts during training. We propose and evaluate two variants of UKGE based on different learning objectives. Experiments are conducted on three real-world uncertain KGs via three tasks, i.e. confidence prediction, relation fact ranking, and relation fact classification. UKGE shows effectiveness in capturing uncertain knowledge by achieving promising results on these tasks, and consistently outperforms baselines on these tasks.

Link prediction for knowledge graphs is the task of predicting missing relationships between entities. Previous work on link prediction has focused on shallow, fast models which can scale to large knowledge graphs. However, these models learn less expressive features than deep, multi-layer models -- which potentially limits performance. In this work, we introduce ConvE, a multi-layer convolutional network model for link prediction, and report state-of-the-art results for several established datasets. We also show that the model is highly parameter efficient, yielding the same performance as DistMult and R-GCN with 8x and 17x fewer parameters. Analysis of our model suggests that it is particularly effective at modelling nodes with high indegree -- which are common in highly-connected, complex knowledge graphs such as Freebase and YAGO3. In addition, it has been noted that the WN18 and FB15k datasets suffer from test set leakage, due to inverse relations from the training set being present in the test set -- however, the extent of this issue has so far not been quantified. We find this problem to be severe: a simple rule-based model can achieve state-of-the-art results on both WN18 and FB15k. To ensure that models are evaluated on datasets where simply exploiting inverse relations cannot yield competitive results, we investigate and validate several commonly used datasets -- deriving robust variants where necessary. We then perform experiments on these robust datasets for our own and several previously proposed models, and find that ConvE achieves state-of-the-art Mean Reciprocal Rank across all datasets.

The dominant sequence transduction models are based on complex recurrent or convolutional neural networks in an encoder-decoder configuration. The best performing models also connect the encoder and decoder through an attention mechanism. We propose a new simple network architecture, the Transformer, based solely on attention mechanisms, dispensing with recurrence and convolutions entirely. Experiments on two machine translation tasks show these models to be superior in quality while being more parallelizable and requiring significantly less time to train. Our model achieves 28.4 BLEU on the WMT 2014 English-to-German translation task, improving over the existing best results, including ensembles by over 2 BLEU. On the WMT 2014 English-to-French translation task, our model establishes a new single-model state-of-the-art BLEU score of 41.8 after training for 3.5 days on eight GPUs, a small fraction of the training costs of the best models from the literature. We show that the Transformer generalizes well to other tasks by applying it successfully to English constituency parsing both with large and limited training data.