Stability of implicit neural networks for long-term forecasting in dynamical systems

Forecasting physical signals in long time range is among the most challenging tasks in Partial Differential Equations (PDEs) research. To circumvent limitations of traditional solvers, many different Deep Learning methods have been proposed. They are all based on auto-regressive methods and exhibit stability issues. Drawing inspiration from the stability property of implicit numerical schemes, we introduce a stable auto-regressive implicit neural network. We develop a theory based on the stability definition of schemes to ensure the stability in forecasting of this network. It leads us to introduce hard constraints on its weights and propagate the dynamics in the latent space. Our experimental results validate our stability property, and show improved results at long-term forecasting for two transports PDEs.Comment: ICLR 2023 Workshop on Physics for Machine Learnin

Functional maps representation on product manifolds

We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices

DS-GPS : A Deep Statistical Graph Poisson Solver (for faster CFD simulations)

This paper proposes a novel Machine Learning-based approach to solve a Poisson problem with mixed boundary conditions. Leveraging Graph Neural Networks, we develop a model able to process unstructured grids with the advantage of enforcing boundary conditions by design. By directly minimizing the residual of the Poisson equation, the model attempts to learn the physics of the problem without the need for exact solutions, in contrast to most previous data-driven processes where the distance with the available solutions is minimized

Reconstruction of turbulent data with deep generative models for semantic inpainting from TURB-Rot database

We study the applicability of tools developed by the computer vision community for features learning and semantic image inpainting to perform data reconstruction of fluid turbulence configurations. The aim is twofold. First, we explore on a quantitative basis, the capability of Convolutional Neural Networks embedded in a Deep Generative Adversarial Model (Deep-GAN) to generate missing data in turbulence, a paradigmatic high dimensional chaotic system. In particular, we investigate their use in reconstructing two-dimensional damaged snapshots extracted from a large database of numerical configurations of 3d turbulence in the presence of rotation, a case with multi-scale random features where both large-scale organised structures and small-scale highly intermittent and non-Gaussian fluctuations are present. Second, following a reverse engineering approach, we aim to rank the input flow properties (features) in terms of their qualitative and quantitative importance to obtain a better set of reconstructed fields. We present two approaches both based on Context Encoders. The first one infers the missing data via a minimization of the L2 pixel-wise reconstruction loss, plus a small adversarial penalisation. The second searches for the closest encoding of the corrupted flow configuration from a previously trained generator. Finally, we present a comparison with a different data assimilation tool, based on Nudging, an equation-informed unbiased protocol, well known in the numerical weather prediction community. The TURB-Rot database, http://smart-turb.roma2.infn.it, of roughly 300K 2d turbulent images is released and details on how to download it are given

Color Recommendation for Vector Graphic Documents based on Multi-Palette Representation

Vector graphic documents present multiple visual elements, such as images, shapes, and texts. Choosing appropriate colors for multiple visual elements is a difficult but crucial task for both amateurs and professional designers. Instead of creating a single color palette for all elements, we extract multiple color palettes from each visual element in a graphic document, and then combine them into a color sequence. We propose a masked color model for color sequence completion and recommend the specified colors based on color context in multi-palette with high probability. We train the model and build a color recommendation system on a large-scale dataset of vector graphic documents. The proposed color recommendation method outperformed other state-of-the-art methods by both quantitative and qualitative evaluations on color prediction and our color recommendation system received positive feedback from professional designers in an interview study.Comment: Accepted to WACV 202

From Data to Knowledge Graphs: A Multi-Layered Method to Model User's Visual Analytics Workflow for Analytical Purposes

The importance of knowledge generation drives much of Visual Analytics (VA). User-tracking and behavior graphs have shown the value of understanding users' knowledge generation while performing VA workflows. Works in theoretical models, ontologies, and provenance analysis have greatly described means to structure and understand the connection between knowledge generation and VA workflows. Yet, two concepts are typically intermixed: the temporal aspect, which indicates sequences of events, and the atemporal aspect, which indicates the workflow state space. In works where these concepts are separated, they do not discuss how to analyze the recorded user's knowledge gathering process when compared to the VA workflow itself. This paper presents Visual Analytic Knowledge Graph (VAKG), a conceptual framework that generalizes existing knowledge models and ontologies by focusing on how humans relate to computer processes temporally and how it relates to the workflow's state space. Our proposal structures this relationship as a 4-way temporal knowledge graph with specific emphasis on modeling the human and computer aspect of VA as separate but interconnected graphs for, among others, analytical purposes. We compare VAKG with relevant literature to show that VAKG's contribution allows VA applications to use it as a provenance model and a state space graph, allowing for analytics of domain-specific processes, usage patterns, and users' knowledge gain performance. We also interviewed two domain experts to check, in the wild, whether real practice and our contributions are aligned.Comment: 9 pgs, submitted to VIS 202