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Over the past few years, neural network methods have evolved in various directions for approximating partial differential equations (PDEs). A promising new development is the integration of neural networks with classical numerical…
This work is devoted to the study of a posteriori error estimation and adaptivity in parabolic problems with a particular focus on spatial discontinuous Galerkin (dG) discretisations. We begin by deriving an a posteriori error estimator for…
Deep learning has become a popular tool across many scientific fields, including the study of differential equations, particularly partial differential equations. This work introduces the basic principles of deep learning and the Deep…
Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be…
Generalized Schr\"odinger Bridges (GSBs) are a fundamental mathematical framework used to analyze the most likely particle evolution based on the principle of least action including kinetic and potential energy. In parallel to their…
We present novel techniques for neuro-symbolic concurrent stochastic games, a recently proposed modelling formalism to represent a set of probabilistic agents operating in a continuous-space environment using a combination of neural network…
We design and analyze reinforcement learning algorithms for Graphon Mean-Field Games (GMFGs). In contrast to previous works that require the precise values of the graphons, we aim to learn the Nash Equilibrium (NE) of the regularized GMFGs…
Improving both accuracy and computational performance of numerical tools is a major challenge for seismic imaging and generally requires specialized implementations to make full use of modern parallel architectures. We present a…
In this paper, we consider discrete-time partially observed mean-field games with the risk-sensitive optimality criterion. We introduce risk-sensitivity behaviour for each agent via an exponential utility function. In the game model, each…
Embedded, or immersed, approaches have the goal of reducing to the minimum the computational costs associated with the generation of body-fitted meshes by only employing fixed, possibly Cartesian, meshes over which complex boundaries can…
Mean field games (MFGs) model equilibria in games with a continuum of weakly interacting players as limiting systems of symmetric $n$-player games. We consider the finite-state, infinite-horizon problem with ergodic cost. Assuming Markovian…
The convexification numerical method with the rigorously established global convergence property is constructed for a problem for the Mean Field Games System of the second order. This is the problem of the retrospective analysis of a game…
This paper studies Mean Field Games (MFGs) in which agent dynamics are given by jump processes of controlled intensity, with mean-field interaction via the controls and affecting the jump intensities. We establish the existence of MFG…
In this paper, we introduce discrete-time linear mean-field games subject to an infinite-horizon discounted-cost optimality criterion. The state space of a generic agent is a compact Borel space. At every time, each agent is randomly…
Deep generative models (DGMs) are effective on learning multilayered representations of complex data and performing inference of input data by exploring the generative ability. However, it is relatively insufficient to empower the…
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering…
Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple…
In this work, we consider a first order mean field games system with non-local couplings. A Lagrange-Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton-Jacobi-Bellman equation, is proposed to…
We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider…
Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as \emph{polytopic}) elements have received considerable attention in recent years. Due to the physical frame basis functions…