Physics-informed attention-based neural network for hyperbolic partial differential equations: application to the Buckley-Leverett problem.

Physics-informed neural networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs) and are in principle capable of modeling a large variety of differential equations. PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that develop highly localized nonlinear shock waves. Learning solutions of PDEs with dominant hyperbolic character is a challenge for current PINN approaches, which rely, like most grid-based numerical schemes, on adding artificial dissipation. Here, we address the fundamental question of which network architectures are best suited to learn the complex behavior of non-linear PDEs. We focus on network architecture rather than on residual regularization. Our new methodology, called physics-informed attention-based neural networks (PIANNs), is a combination of recurrent neural networks and attention mechanisms. The attention mechanism adapts the behavior of the deep neural network to the non-linear features of the solution, and break the current limitations of PINNs. We find that PIANNs effectively capture the shock front in a hyperbolic model problem, and are capable of providing high-quality solutions inside the convex hull of the training set.

Playing Atari with few neurons: Improving the efficacy of reinforcement learning by decoupling feature extraction and decision making.

We propose a new method for learning compact state representations and policies separately but simultaneously for policy approximation in vision-based applications such as Atari games. Approaches based on deep reinforcement learning typically map pixels directly to actions to enable end-to-end training. Internally, however, the deep neural network bears the responsibility of both extracting useful information and making decisions based on it, two objectives which can be addressed independently. Separating the image processing from the action selection allows for a better understanding of either task individually, as well as potentially finding smaller policy representations which is inherently interesting. Our approach learns state representations using a compact encoder based on two novel algorithms: (i) Increasing Dictionary Vector Quantization builds a dictionary of state representations which grows in size over time, allowing our method to address new observations as they appear in an open-ended online-learning context; and (ii) Direct Residuals Sparse Coding encodes observations in function of the dictionary, aiming for highest information inclusion by disregarding reconstruction error and maximizing code sparsity. As the dictionary size increases, however, the encoder produces increasingly larger inputs for the neural network; this issue is addressed with a new variant of the Exponential Natural Evolution Strategies algorithm which adapts the dimensionality of its probability distribution along the run. We test our system on a selection of Atari games using tiny neural networks of only 6 to 18 neurons (depending on each game's controls). These are still capable of achieving results that are not much worse, and occasionally superior, to the state-of-the-art in direct policy search which uses two orders of magnitude more neurons.

Constrained novelty search: a study on game content generation.

Novelty search is a recent algorithm geared toward exploring search spaces without regard to objectives. When the presence of constraints divides a search space into feasible space and infeasible space, interesting implications arise regarding how novelty search explores such spaces. This paper elaborates on the problem of constrained novelty search and proposes two novelty search algorithms which search within both the feasible and the infeasible space. Inspired by the FI-2pop genetic algorithm, both algorithms maintain and evolve two separate populations, one with feasible and one with infeasible individuals, while each population can use its own selection method. The proposed algorithms are applied to the problem of generating diverse but playable game levels, which is representative of the larger problem of procedural game content generation. Results show that the two-population constrained novelty search methods can create, under certain conditions, larger and more diverse sets of feasible game levels than current methods of novelty search, whether constrained or unconstrained. However, the best algorithm is contingent on the particularities of the search space and the genetic operators used. Additionally, the proposed enhancement of offspring boosting is shown to enhance performance in all cases of two-population novelty search.