Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an emerging research field. However, current methods are restricted in various ways: they require prior knowledge about the governing equations, and are limited to linear or first-order equations. In this work we propose NeuralPDE, a model which combines convolutional neural networks (CNNs) with differentiable ODE solvers to model dynamical systems. We show that the Method of Lines used in standard PDE solvers can be represented using convolutions which makes CNNs the natural choice to parametrize arbitrary PDE dynamics. Our model can be applied to any data without requiring any prior knowledge about the governing PDE. We evaluate NeuralPDE on datasets generated by solving a wide variety of PDEs, covering higher orders, non-linear equations and multiple spatial dimensions.
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Having a model and being able to implement open-ended evolutionary systems is important for advancing our understanding of open-endedness. Complex systems science and newest generation high-level programming languages provide intriguing possibilities to do so, respectively. Here, some recent advances in modelling and implementing open-ended evolutionary systems are reviewed first. Then, the so-called allagmatic method to describe, model, implement, and interpret complex systems is introduced. After highlighting some current modelling and implementation challenges, model building blocks of open-ended evolutionary systems are identified, a system metamodel of open-ended evolution is formalised in the allagmatic method, and an implementation prototype with a high-level programming language is outlined. The proposed approach shows statistical characteristics of open-ended evolutionary systems and provides a promising starting point to interpret novelty generated at runtime.
Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms
Motivated by applications in single-cell biology and metagenomics, we consider matrix reordering based on the noisy disordered matrix model. We first establish the fundamental statistical limit for the matrix reordering problem in a decision-theoretic framework and show that a constrained least square estimator is rate-optimal. Given the computational hardness of the optimal procedure, we analyze a popular polynomial-time algorithm, spectral seriation, and show that it is suboptimal. We then propose a novel polynomial-time adaptive sorting algorithm with guaranteed improvement on the performance. The superiority of the adaptive sorting algorithm over the existing methods is demonstrated in simulation studies and in the analysis of two real single-cell RNA sequencing datasets.
Analytics corresponds to a relevant and challenging phase of Big Data. The generation of knowledge from extensive data sets (petabyte era) of varying types, occurring at a speed able to serve decision makers, is practiced using multiple areas of knowledge, such as computing, statistics, data mining, among others. In the Big Data domain, Analytics is also considered as a process capable of adding value to the organizations. Besides the demonstration of value, Analytics should also consider operational tools and models to support decision making. To adding value, Analytics is also presented as part of some Big Data value chains, such the Information Value Chain presented by NIST among others, which are detailed in this article. As well, some maturity models are presented, since they represent important structures to favor continuous implementation of Analytics for Big Data, using specific technologies, techniques and methods. Hence, through an in-depth research, using specific literature references and use cases, we seeks to outline an approach to determine the Analytical Engineering for Big Data Analytics considering four pillars: Data, Models, Tools and People; and three process groups: Acquisition, Retention and Revision; in order to make feasible and to define an organization, possibly designated as an Analytics Organization, responsible for generating knowledge from the data in the field of Big Data Analytics.
The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do {\em not} need to be observed or approximated and {\em no} prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.
Partial Differential Equations (PDEs) describe several problems relevant to many fields of applied sciences, and their discrete counterparts typically involve the solution of sparse linear systems. In this context, we focus on the analysis of the computational aspects related to the solution of large and sparse linear systems with HPC solvers, by considering the performances of direct and iterative solvers in terms of computational efficiency, scalability, and numerical accuracy. Our aim is to identify the main criteria to support application-domain specialists in the selection of the most suitable solvers, according to the application requirements and available resources. To this end, we discuss how the numerical solver is affected by the regular/irregular discretisation of the input domain, the discretisation of the input PDE with piecewise linear or polynomial basis functions, which generally result in a higher/lower sparsity of the coefficient matrix, and the choice of different initial conditions, which are associated with linear systems with multiple right-hand side terms. Finally, our analysis is independent of the characteristics of the underlying computational architectures, and provides a methodological approach that can be applied to different classes of PDEs or with approximation problems.
We study constrained reinforcement learning (CRL) from a novel perspective by setting constraints directly on state density functions, rather than the value functions considered by previous works. State density has a clear physical and mathematical interpretation, and is able to express a wide variety of constraints such as resource limits and safety requirements. Density constraints can also avoid the time-consuming process of designing and tuning cost functions required by value function-based constraints to encode system specifications. We leverage the duality between density functions and Q functions to develop an effective algorithm to solve the density constrained RL problem optimally and the constrains are guaranteed to be satisfied. We prove that the proposed algorithm converges to a near-optimal solution with a bounded error even when the policy update is imperfect. We use a set of comprehensive experiments to demonstrate the advantages of our approach over state-of-the-art CRL methods, with a wide range of density constrained tasks as well as standard CRL benchmarks such as Safety-Gym.
Promoting behavioural diversity is critical for solving games with non-transitive dynamics where strategic cycles exist, and there is no consistent winner (e.g., Rock-Paper-Scissors). Yet, there is a lack of rigorous treatment for defining diversity and constructing diversity-aware learning dynamics. In this work, we offer a geometric interpretation of behavioural diversity in games and introduce a novel diversity metric based on \emph{determinantal point processes} (DPP). By incorporating the diversity metric into best-response dynamics, we develop \emph{diverse fictitious play} and \emph{diverse policy-space response oracle} for solving normal-form games and open-ended games. We prove the uniqueness of the diverse best response and the convergence of our algorithms on two-player games. Importantly, we show that maximising the DPP-based diversity metric guarantees to enlarge the \emph{gamescape} -- convex polytopes spanned by agents' mixtures of strategies. To validate our diversity-aware solvers, we test on tens of games that show strong non-transitivity. Results suggest that our methods achieve much lower exploitability than state-of-the-art solvers by finding effective and diverse strategies.
Deep Learning is applied to energy markets to predict extreme loads observed in energy grids. Forecasting energy loads and prices is challenging due to sharp peaks and troughs that arise due to supply and demand fluctuations from intraday system constraints. We propose deep spatio-temporal models and extreme value theory (EVT) to capture theses effects and in particular the tail behavior of load spikes. Deep LSTM architectures with ReLU and $\tanh$ activation functions can model trends and temporal dependencies while EVT captures highly volatile load spikes above a pre-specified threshold. To illustrate our methodology, we use hourly price and demand data from 4719 nodes of the PJM interconnection, and we construct a deep predictor. We show that DL-EVT outperforms traditional Fourier time series methods, both in-and out-of-sample, by capturing the observed nonlinearities in prices. Finally, we conclude with directions for future research.
Nowadays, recommender systems are present in many daily activities such as online shopping, browsing social networks, etc. Given the rising demand for reinvigoration of the tourist industry through information technology, recommenders have been included into tourism websites such as Expedia, Booking or Tripadvisor, among others. Furthermore, the amount of scientific papers related to recommender systems for tourism is on solid and continuous growth since 2004. Much of this growth is due to social networks that, besides to offer researchers the possibility of using a great mass of available and constantly updated data, they also enable the recommendation systems to become more personalised, effective and natural. This paper reviews and analyses many research publications focusing on tourism recommender systems that use social networks in their projects. We detail their main characteristics, like which social networks are exploited, which data is extracted, the applied recommendation techniques, the methods of evaluation, etc. Through a comprehensive literature review, we aim to collaborate with the future recommender systems, by giving some clear classifications and descriptions of the current tourism recommender systems.
Most existing recommender systems leverage user behavior data of one type only, such as the purchase behavior in E-commerce that is directly related to the business KPI (Key Performance Indicator) of conversion rate. Besides the key behavioral data, we argue that other forms of user behaviors also provide valuable signal, such as views, clicks, adding a product to shop carts and so on. They should be taken into account properly to provide quality recommendation for users. In this work, we contribute a new solution named NMTR (short for Neural Multi-Task Recommendation) for learning recommender systems from user multi-behavior data. We develop a neural network model to capture the complicated and multi-type interactions between users and items. In particular, our model accounts for the cascading relationship among different types of behaviors (e.g., a user must click on a product before purchasing it). To fully exploit the signal in the data of multiple types of behaviors, we perform a joint optimization based on the multi-task learning framework, where the optimization on a behavior is treated as a task. Extensive experiments on two real-world datasets demonstrate that NMTR significantly outperforms state-of-the-art recommender systems that are designed to learn from both single-behavior data and multi-behavior data. Further analysis shows that modeling multiple behaviors is particularly useful for providing recommendation for sparse users that have very few interactions.
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