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We present the application of the physics-informed neural network (PINN) approach in Bayesian formulation. We have adopted the Bayesian neural network framework to obtain posterior densities from Laplace approximation. For each model or fit, the evidence is computed, which is a measure that classifies the hypothesis. The optimal solution is the one with the highest value of evidence. We have proposed a modification of the Bayesian algorithm to obtain hyperparameters of the model. We have shown that within the Bayesian framework, one can obtain the relative weights between the boundary and equation contributions to the total loss. Presented method leads to predictions comparable to those obtained by sampling from the posterior distribution within the Hybrid Monte Carlo algorithm (HMC). We have solved heat, wave, and Burger's equations, and the results obtained are in agreement with the exact solutions, demonstrating the effectiveness of our approach. In Burger's equation problem, we have demonstrated that the framework can combine information from differential equations and potential measurements. All solutions are provided with uncertainties (induced by the model's parameter dependence) computed within the Bayesian framework.

Data driven modelling and scientific machine learning have been responsible for significant advances in determining suitable models to describe data. Within dynamical systems, neural ordinary differential equations (ODEs), where the system equations are set to be governed by a neural network, have become a popular tool for this challenge in recent years. However, less emphasis has been placed on systems that are only partially-observed. In this work, we employ a hybrid neural ODE structure, where the system equations are governed by a combination of a neural network and domain-specific knowledge, together with symbolic regression (SR), to learn governing equations of partially-observed dynamical systems. We test this approach on two case studies: A 3-dimensional model of the Lotka-Volterra system and a 5-dimensional model of the Lorenz system. We demonstrate that the method is capable of successfully learning the true underlying governing equations of unobserved states within these systems, with robustness to measurement noise.

We propose a scheme leveraging reinforcement learning to engineer control fields for generating non-classical states. It is exemplified by the application to prepare spin-squeezed states for an open collective spin model where a linear control field is designed to govern the dynamics. The reinforcement learning agent determines the temporal sequence of control pulses, commencing from a coherent spin state in an environment characterized by dissipation and dephasing. Compared to the constant control scenario, this approach provides various control sequences maintaining collective spin squeezing and entanglement. It is observed that denser application of the control pulses enhances the performanceof the outcomes. However, there is a minor enhancement in the performance by adding control actions. The proposed strategy demonstrates increased effectiveness for larger systems. Thermal excitations of the reservoir are detrimental to the control outcomes. Feasible experiments are suggested to implement the control proposal. The extension to continuous control problems and another quantum system are discussed. The replaceability of the reinforcement learning module is also emphasized. This research paves the way for its application in manipulating other quantum systems.

We present a new methodology for utilising machine learning technology in symbolic computation research. We explain how a well known human-designed heuristic to make the choice of variable ordering in cylindrical algebraic decomposition may be represented as a constrained neural network. This allows us to then use machine learning methods to further optimise the heuristic, leading to new networks of similar size, representing new heuristics of similar complexity as the original human-designed one. We present this as a form of ante-hoc explainability for use in computer algebra development.

Causality can be described in terms of a structural causal model (SCM) that carries information on the variables of interest and their mechanistic relations. For most processes of interest the underlying SCM will only be partially observable, thus causal inference tries to leverage any exposed information. Graph neural networks (GNN) as universal approximators on structured input pose a viable candidate for causal learning, suggesting a tighter integration with SCM. To this effect we present a theoretical analysis from first principles that establishes a novel connection between GNN and SCM while providing an extended view on general neural-causal models. We then establish a new model class for GNN-based causal inference that is necessary and sufficient for causal effect identification. Our empirical illustration on simulations and standard benchmarks validate our theoretical proofs.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.

There has been appreciable progress in unsupervised network representation learning (UNRL) approaches over graphs recently with flexible random-walk approaches, new optimization objectives and deep architectures. However, there is no common ground for systematic comparison of embeddings to understand their behavior for different graphs and tasks. In this paper we theoretically group different approaches under a unifying framework and empirically investigate the effectiveness of different network representation methods. In particular, we argue that most of the UNRL approaches either explicitly or implicit model and exploit context information of a node. Consequently, we propose a framework that casts a variety of approaches -- random walk based, matrix factorization and deep learning based -- into a unified context-based optimization function. We systematically group the methods based on their similarities and differences. We study the differences among these methods in detail which we later use to explain their performance differences (on downstream tasks). We conduct a large-scale empirical study considering 9 popular and recent UNRL techniques and 11 real-world datasets with varying structural properties and two common tasks -- node classification and link prediction. We find that there is no single method that is a clear winner and that the choice of a suitable method is dictated by certain properties of the embedding methods, task and structural properties of the underlying graph. In addition we also report the common pitfalls in evaluation of UNRL methods and come up with suggestions for experimental design and interpretation of results.

Embedding entities and relations into a continuous multi-dimensional vector space have become the dominant method for knowledge graph embedding in representation learning. However, most existing models ignore to represent hierarchical knowledge, such as the similarities and dissimilarities of entities in one domain. We proposed to learn a Domain Representations over existing knowledge graph embedding models, such that entities that have similar attributes are organized into the same domain. Such hierarchical knowledge of domains can give further evidence in link prediction. Experimental results show that domain embeddings give a significant improvement over the most recent state-of-art baseline knowledge graph embedding models.

We advocate the use of implicit fields for learning generative models of shapes and introduce an implicit field decoder for shape generation, aimed at improving the visual quality of the generated shapes. An implicit field assigns a value to each point in 3D space, so that a shape can be extracted as an iso-surface. Our implicit field decoder is trained to perform this assignment by means of a binary classifier. Specifically, it takes a point coordinate, along with a feature vector encoding a shape, and outputs a value which indicates whether the point is outside the shape or not. By replacing conventional decoders by our decoder for representation learning and generative modeling of shapes, we demonstrate superior results for tasks such as shape autoencoding, generation, interpolation, and single-view 3D reconstruction, particularly in terms of visual quality.

Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.