Fabricio Arend Torres

Fabricio Arend Torres, Marcello Massimo Negri, Monika Nagy-Huber et al.

Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free, current approaches still rely on collocation points within a bounded region, even in settings with spatially sparse signals. Furthermore, if the boundaries are not known, the selection of such a region is difficult and often results in a large proportion of collocation points being selected in areas of low relevance. To resolve this severe drawback of current methods, we present a mesh-free and adaptive approach termed particle-density PINN (pdPINN), which is inspired by the microscopic viewpoint of fluid dynamics. The method is based on the Eulerian formulation and, different from classical mesh-free method, does not require the introduction of Lagrangian updates. We propose to sample directly from the distribution over the particle positions, eliminating the need to introduce boundaries while adaptively focusing on the most relevant regions. This is achieved by interpreting a non-negative physical quantity (such as the density or temperature) as an unnormalized probability distribution from which we sample with dynamic Monte Carlo methods. The proposed method leads to higher sample efficiency and improved performance of PINNs. These advantages are demonstrated on various experiments based on the continuity equations, Fokker-Planck equations, and the heat equation.

Vitali Nesterov, Fabricio Arend Torres, Monika Nagy-Huber et al.

Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level sets. Specifically, this characterisation should be of a global parametric form, which is especially useful for different informed data exploration tasks, such as building grid-based approximations, sampling points along the level curves, or finding trajectories on the manifold. However, global parameterisations can only exist if the level sets are connected. For this purpose, we introduce a novel and flexible class of neural networks that generalise input-convex networks. These networks represent functions that are guaranteed to have connected level sets forming smooth manifolds on the input space. We further show that global parameterisations of these level sets can be always found efficiently. Lastly, we demonstrate that our novel technique for characterising invariances is a powerful generative data exploration tool in real-world applications, such as computational chemistry.

Sebastian Mathias Keller, Maxim Samarin, Fabricio Arend Torres et al.

Archetypes are typical population representatives in an extremal sense, where typicality is understood as the most extreme manifestation of a trait or feature. In linear feature space, archetypes approximate the data convex hull allowing all data points to be expressed as convex mixtures of archetypes. However, it might not always be possible to identify meaningful archetypes in a given feature space. Learning an appropriate feature space and identifying suitable archetypes simultaneously addresses this problem. This paper introduces a generative formulation of the linear archetype model, parameterized by neural networks. By introducing the distance-dependent archetype loss, the linear archetype model can be integrated into the latent space of a variational autoencoder, and an optimal representation with respect to the unknown archetypes can be learned end-to-end. The reformulation of linear Archetypal Analysis as deep variational information bottleneck, allows the incorporation of arbitrarily complex side information during training. Furthermore, an alternative prior, based on a modified Dirichlet distribution, is proposed. The real-world applicability of the proposed method is demonstrated by exploring archetypes of female facial expressions while using multi-rater based emotion scores of these expressions as side information. A second application illustrates the exploration of the chemical space of small organic molecules. In this experiment, it is demonstrated that exchanging the side information but keeping the same set of molecules, e. g. using as side information the heat capacity of each molecule instead of the band gap energy, will result in the identification of different archetypes. As an application, these learned representations of chemical space might reveal distinct starting points for de novo molecular design.