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Critical points mark locations in the domain where the level-set topology of a scalar function undergoes fundamental changes and thus indicate potentially interesting features in the data. Established methods exist to locate and relate such points in a deterministic setting, but it is less well understood how the concept of critical points can be extended to the analysis of uncertain data. Most methods for this task aim at finding likely locations of critical points or estimate the probability of their occurrence locally but do not indicate if critical points at potentially different locations in different realizations of a stochastic process are manifestations of the same feature, which is required to characterize the spatial uncertainty of critical points. Previous work on relating critical points across different realizations reported challenges for interpreting the resulting spatial distribution of critical points but did not investigate the causes. In this work, we provide a mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution, which leads us to the notion of uncertain critical points. We analyze the theoretical properties of these structures and highlight connections to existing works for special classes of uncertain fields. We derive conditions under which well-interpretable results can be obtained and discuss the implications of those restrictions for the field of visualization. We demonstrate that the discussed limitations are not purely academic but also arise in real-world data.

We present a novel form of Fourier analysis, and associated signal processing concepts, for signals (or data) indexed by edge-weighted directed acyclic graphs (DAGs). This means that our Fourier basis yields an eigendecomposition of a suitable notion of shift and convolution operators that we define. DAGs are the common model to capture causal relationships between data values and in this case our proposed Fourier analysis relates data with its causes under a linearity assumption that we define. The definition of the Fourier transform requires the transitive closure of the weighted DAG for which several forms are possible depending on the interpretation of the edge weights. Examples include level of influence, distance, or pollution distribution. Our framework is different from prior GSP: it is specific to DAGs and leverages, and extends, the classical theory of Moebius inversion from combinatorics. For a prototypical application we consider DAGs modeling dynamic networks in which edges change over time. Specifically, we model the spread of an infection on such a DAG obtained from real-world contact tracing data and learn the infection signal from samples assuming sparsity in the Fourier domain.

Novel view synthesis and 3D modeling using implicit neural field representation are shown to be very effective for calibrated multi-view cameras. Such representations are known to benefit from additional geometric and semantic supervision. Most existing methods that exploit additional supervision require dense pixel-wise labels or localized scene priors. These methods cannot benefit from high-level vague scene priors provided in terms of scenes' descriptions. In this work, we aim to leverage the geometric prior of Manhattan scenes to improve the implicit neural radiance field representations. More precisely, we assume that only the knowledge of the indoor scene (under investigation) being Manhattan is known -- with no additional information whatsoever -- with an unknown Manhattan coordinate frame. Such high-level prior is used to self-supervise the surface normals derived explicitly in the implicit neural fields. Our modeling allows us to cluster the derived normals and exploit their orthogonality constraints for self-supervision. Our exhaustive experiments on datasets of diverse indoor scenes demonstrate the significant benefit of the proposed method over the established baselines. The source code will be available at //github.com/nikola3794/normal-clustering-nerf.

Even though the analysis of unsteady 2D flow fields is challenging, fluid mechanics experts generally have an intuition on where in the simulation domain specific features are expected. Using this intuition, showing similar regions enables the user to discover flow patterns within the simulation data. When focusing on similarity, a solid mathematical framework for a specific flow pattern is not required. We propose a technique that visualizes similar and dissimilar regions with respect to a region selected by the user. Using infinitesimal strain theory, we capture the strain and rotation progression and therefore the dynamics of fluid parcels along pathlines, which we encode as distributions. We then apply the Jensen-Shannon divergence to compute the (dis)similarity between pathline dynamics originating in a user-defined flow region and the pathline dynamics of the flow field. We validate our method by applying it to two simulation datasets of two-dimensional unsteady flows. Our results show that our approach is suitable for analyzing the similarity of time-dependent flow fields.

The curse-of-dimensionality (CoD) taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs as Richard Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. In this paper, we develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and samples randomly a subset of these dimensional pieces in each iteration of training PINNs. We theoretically prove the convergence guarantee and other desired properties of the proposed method. We experimentally demonstrate that the proposed method allows us to solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schr\"{o}dinger equations in thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. For instance, we solve nontrivial nonlinear PDEs (one HJB equation and one Black-Scholes equation) in 100,000 dimensions in 6 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, SDGD can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

DAVIS camera, streaming two complementary sensing modalities of asynchronous events and frames, has gradually been used to address major object detection challenges (e.g., fast motion blur and low-light). However, how to effectively leverage rich temporal cues and fuse two heterogeneous visual streams remains a challenging endeavor. To address this challenge, we propose a novel streaming object detector with Transformer, namely SODFormer, which first integrates events and frames to continuously detect objects in an asynchronous manner. Technically, we first build a large-scale multimodal neuromorphic object detection dataset (i.e., PKU-DAVIS-SOD) over 1080.1k manual labels. Then, we design a spatiotemporal Transformer architecture to detect objects via an end-to-end sequence prediction problem, where the novel temporal Transformer module leverages rich temporal cues from two visual streams to improve the detection performance. Finally, an asynchronous attention-based fusion module is proposed to integrate two heterogeneous sensing modalities and take complementary advantages from each end, which can be queried at any time to locate objects and break through the limited output frequency from synchronized frame-based fusion strategies. The results show that the proposed SODFormer outperforms four state-of-the-art methods and our eight baselines by a significant margin. We also show that our unifying framework works well even in cases where the conventional frame-based camera fails, e.g., high-speed motion and low-light conditions. Our dataset and code can be available at //github.com/dianzl/SODFormer.

Recently, graph neural networks have been gaining a lot of attention to simulate dynamical systems due to their inductive nature leading to zero-shot generalizability. Similarly, physics-informed inductive biases in deep-learning frameworks have been shown to give superior performance in learning the dynamics of physical systems. There is a growing volume of literature that attempts to combine these two approaches. Here, we evaluate the performance of thirteen different graph neural networks, namely, Hamiltonian and Lagrangian graph neural networks, graph neural ODE, and their variants with explicit constraints and different architectures. We briefly explain the theoretical formulation highlighting the similarities and differences in the inductive biases and graph architecture of these systems. We evaluate these models on spring, pendulum, gravitational, and 3D deformable solid systems to compare the performance in terms of rollout error, conserved quantities such as energy and momentum, and generalizability to unseen system sizes. Our study demonstrates that GNNs with additional inductive biases, such as explicit constraints and decoupling of kinetic and potential energies, exhibit significantly enhanced performance. Further, all the physics-informed GNNs exhibit zero-shot generalizability to system sizes an order of magnitude larger than the training system, thus providing a promising route to simulate large-scale realistic systems.

Graphs are used widely to model complex systems, and detecting anomalies in a graph is an important task in the analysis of complex systems. Graph anomalies are patterns in a graph that do not conform to normal patterns expected of the attributes and/or structures of the graph. In recent years, graph neural networks (GNNs) have been studied extensively and have successfully performed difficult machine learning tasks in node classification, link prediction, and graph classification thanks to the highly expressive capability via message passing in effectively learning graph representations. To solve the graph anomaly detection problem, GNN-based methods leverage information about the graph attributes (or features) and/or structures to learn to score anomalies appropriately. In this survey, we review the recent advances made in detecting graph anomalies using GNN models. Specifically, we summarize GNN-based methods according to the graph type (i.e., static and dynamic), the anomaly type (i.e., node, edge, subgraph, and whole graph), and the network architecture (e.g., graph autoencoder, graph convolutional network). To the best of our knowledge, this survey is the first comprehensive review of graph anomaly detection methods based on GNNs.

The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.

Graph Convolutional Networks (GCNs) have been widely applied in various fields due to their significant power on processing graph-structured data. Typical GCN and its variants work under a homophily assumption (i.e., nodes with same class are prone to connect to each other), while ignoring the heterophily which exists in many real-world networks (i.e., nodes with different classes tend to form edges). Existing methods deal with heterophily by mainly aggregating higher-order neighborhoods or combing the immediate representations, which leads to noise and irrelevant information in the result. But these methods did not change the propagation mechanism which works under homophily assumption (that is a fundamental part of GCNs). This makes it difficult to distinguish the representation of nodes from different classes. To address this problem, in this paper we design a novel propagation mechanism, which can automatically change the propagation and aggregation process according to homophily or heterophily between node pairs. To adaptively learn the propagation process, we introduce two measurements of homophily degree between node pairs, which is learned based on topological and attribute information, respectively. Then we incorporate the learnable homophily degree into the graph convolution framework, which is trained in an end-to-end schema, enabling it to go beyond the assumption of homophily. More importantly, we theoretically prove that our model can constrain the similarity of representations between nodes according to their homophily degree. Experiments on seven real-world datasets demonstrate that this new approach outperforms the state-of-the-art methods under heterophily or low homophily, and gains competitive performance under homophily.