Diogo A. Gomes

Noha Almulla, Rita Ferreira, Diogo Gomes

Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including one-dimensional periodic MFG, congestion problems, and higher-dimensional models.

Diogo Gomes, Joao Saude

Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions are the main difficulty in the numerical approximation of solutions. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve the MFG, both for stationary and time-dependent problems. We illustrate our methods in a MFG modeling the paradigm-shift problem.

Diogo A. Gomes, Joana Mohr, Rafael R. Souza

Mean field games is a recent area of study introduced by Lions and Lasry in a series of seminal papers in 2006. Mean field games model situations of competition between large number of rational agents that play non-cooperative dynamic games under certain symmetry assumptions. They key step is to develop a mean field model, in a similar way that what is done in statistical physics in order to construct a mathematically tractable model. A main question that arises in the study of such mean field problems is the rigorous justification of the mean field models by a limiting procedure. In this paper we consider the mean field limit of two-state Markov decision problem as the number of players $N\to \infty$. First we establish the existence and uniqueness of a symmetric partial information Markov perfect equilibrium. Then we derive a mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. Our main result is the convergence as $N\to \infty$ of the $N$ player game to the mean field model and an estimate of the rate of convergence.

Filippo Cagnetti, Diogo Gomes, Hung V. Tran

We consider a numerical scheme for the one dimensional time dependent Hamilton-Jacobi equation in the periodic setting. We present a new and simple proof of the rate of convergence of the approximations based on the adjoint method recently introduced by Evans.

Filipe Barroso, Diogo Gomes, Gareth J. Baxter

We propose a constraint-based algorithm, which automatically determines causal relevance thresholds, to infer causal networks from data. We call these topological thresholds. We present two methods for determining the threshold: the first seeks a set of edges that leaves no disconnected nodes in the network; the second seeks a causal large connected component in the data. We tested these methods both for discrete synthetic and real data, and compared the results with those obtained for the PC algorithm, which we took as the benchmark. We show that this novel algorithm is generally faster and more accurate than the PC algorithm. The algorithm for determining the thresholds requires choosing a measure of causality. We tested our methods for Fisher Correlations, commonly used in PC algorithm (for instance in \cite{kalisch2005}), and further proposed a discrete and asymmetric measure of causality, that we called Net Influence, which provided very good results when inferring causal networks from discrete data. This metric allows for inferring directionality of the edges in the process of applying the thresholds, speeding up the inference of causal DAGs.

Diogo Gomes, Alexandre Evsukoff

Over the last few decades, institutions around the world have been challenged to deal with the sheer volume of information captured in unstructured formats, especially in textual documents. The so called Digital Transformation age, characterized by important technological advances and the advent of disruptive methods in Artificial Intelligence, offers opportunities to make better use of this information. Recent techniques in Natural Language Processing (NLP) with Deep Learning approaches allow to efficiently process a large volume of data in order to obtain relevant information, to identify patterns, classify text, among other applications. In this context, the highly technical vocabulary of Oil and Gas (O&G) domain represents a challenge for these NLP algorithms, in which terms can assume a very different meaning in relation to common sense understanding. The search for suitable mathematical representations and specific models requires a large amount of representative corpora in the O&G domain. However, public access to this material is scarce in the scientific literature, especially considering the Portuguese language. This paper presents a literature review about the main techniques for deep learning NLP and their major applications for O&G domain in Portuguese.