President Trump announces the U.S. Stargate investment alongside three artificial intelligence industry leaders. President Donald Trump's high-tech moonshot may hit a Texas-sized speed bump -- and it's coming from his own party. Trump's AI initiative, dubbed "Stargate," aims to build 20 ultra-powerful data centers across the country. Backed by heavyweights like OpenAI, Oracle, SoftBank, and the UAE-funded MGX, the project represents a 500 billion bet on the future with Texas chosen as ground zero for the first 10 centers. But a new Texas bill, Senate Bill 6, could delay or derail that momentum.

Geometry is a ubiquitous language of computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) to train shape generative models \emph{without any data}. GINNs combine (i) learning under constraints, (ii) neural fields as a suitable representation, and (iii) generating diverse solutions to under-determined problems. We apply GINNs to several two and three-dimensional problems of increasing levels of complexity. Our results demonstrate the feasibility of training shape generative models in a data-free setting. This new paradigm opens several exciting research directions, expanding the application of generative models into domains where data is sparse.

Graph Neural Networks (GNNs) are well known as powerful tools for learning tasks on graphstructured data [10], such as semi-supervised node classification, link prediction, and graph classification, with their origin that dates back to the late 2000s [4, 6, 7]. Recently, a new type of layer for GNNs called Graph-Informed (GI) layer [1] has been developed, specifically designed for regression tasks on graph-nodes; indeed, this type of task is not suitable for classic GNNs and, therefore, typically it is approached using MLPs, that do not exploit the graph structure of the data. Nonetheless, the usage of GI layers has been recently extended also to supervised classification tasks (see [3]). The main advantages of the GI layers is the possibility to build Neural Networks (NNs), called Graph-Informed NNs (GINNs), suitable for large graphs and deep architectures. Their good performances, especially if compared with respect to classic MLPs, are illustrated both in [1] (regression tasks) and [3] (classification task for discontinuity detection). However, at the time this work is written, existing GI layer implementations have one main limitation.

Detecting discontinuity interfaces of discontinuous functions is a challenging task with significant implications across various scientific and engineering applications. Identifying these interfaces is particularly critical for functions with a high-dimensional domain, as their discontinuities can significantly influence the behavior of numerical methods and simulations; for example, within the realm of uncertainty quantification, where the smoothness of the target function plays a fundamental role in the use of stochastic collocation methods. Specifically, the knowledge of discontinuity interfaces enables the partitioning of the function domain into regions of smoothness, a crucial factor in improving the performance of numerical methods (e.g., see [17]). Other examples of discontinuity detection applications include signal processing, nonlinear partial differential equation (PDE) simulations, investigations of phase transitions in physical systems [14], and change-point analyses in geology or biology, to name a few [30]. The central objective of most discontinuity detection methods is to identify the position of discontinuities in the function domain using function evaluations on sets of points. Over the last few decades, progresses has been made in discontinuity detection, leading to the development of various algorithms. Notable works, such as [3, 2, 16, 35], have introduced significant contributions in this field. In particular, [3] introduced a polynomial annihilation edge detection method designed for piece-wise smooth functions with low-dimensional domains (n 2). This method identifies discontinuous interfaces by reconstructing jump functions based on a set of function evaluations.

Timely completion of design cycles for multiscale and multiphysics systems ranging from consumer electronics to hypersonic vehicles relies on rapid simulation-based prototyping. The latter typically involves high-dimensional spaces of possibly correlated control variables (CVs) and quantities of interest (QoIs) with non-Gaussian and/or multimodal distributions. We develop a model-agnostic, moment-independent global sensitivity analysis (GSA) that relies on differential mutual information to rank the effects of CVs on QoIs. Large amounts of data, which are necessary to rank CVs with confidence, are cheaply generated by a deep neural network (DNN) surrogate model of the underlying process. The DNN predictions are made explainable by the GSA so that the DNN can be deployed to close design loops. Our information-theoretic framework is compatible with a wide variety of black-box models. Its application to multiscale supercapacitor design demonstrates that the CV rankings facilitated by a domain-aware Graph-Informed Neural Network are better resolved than their counterparts obtained with a physics-based model for a fixed computational budget. Consequently, our information-theoretic GSA provides an "outer loop" for accelerated product design by identifying the most and least sensitive input directions and performing subsequent optimization over appropriately reduced parameter subspaces.

Typically this requires casting the original deterministic physics-based model into a probabilistic framework where inputs or control variables (CVs) are treated as random variables with probability distributions derived from available experimental data, manufacturing constraints, design criteria, expert judgment, and/or other domain knowledge (e.g., see [1]). Running the physics-based model with CVs sampled according to these distributions yields corresponding realizations of the system response as characterized by quantities of interest (QoIs). Analysis of the uncertainty propagation from the CVs to the QoIs informs decision-making, e.g., it informs engineering decisions aimed at improving the quality and reliability of designed products and helps identify potential risks at early stages in the design and manufacturing process. Quantitatively assessing uncertainty propagation presents a fundamental challenge due to the computational cost of the underlying physics-based model. Even for a low number of CVs and QoIs, uncertainty quantification (UQ) for, e.g., accelerating the simulation-aided design of multiscale systems and data-centric engineering tasks more generally ([2]), requires a large number of repeated observations of QoIs to achieve a high degree of confidence in such an analysis. The sampling cost is further exacerbated in real-world applications where distributions on QoIs are typically non-Gaussian, skewed, and/or mutually correlated, and therefore need to be characterized by their full probability density function (PDF) rather than through summary statistics such as mean and variance. The computational cost of nonparametric methods to estimate these densities can become prohibitively high when using a fully-featured physics-based model to compute each sample. One approach to alleviate the computational burden is to derive a cheaper-to-compute surrogate for the physicsbased model's response enabling much faster generation of output data and thus overcoming computational bottlenecks.

This informal technical report details the geometric illustration of decision boundaries for ReLU units in a three layer fully connected neural network. The network is designed and trained to predict pixel intensity from an (x, y) input location. The Geometric Illustration of Neural Networks (GINN) tool was built to visualise and track the points at which ReLU units switch from being active to off (or vice versa) as the network undergoes training. Several phenomenon were observed and are discussed herein. This technical report is a supporting document to the blog post with online demos and is available at http://www.bayeswatch.com/2018/09/17/GINN/.