The numerical solution of partial differential equations (PDEs) is difficult, having led to a century of research so far. Recently, there have been pushes to build neural--numerical hybrid solvers, which piggy-backs the modern trend towards fully end-to-end learned systems. Most works so far can only generalize over a subset of properties to which a generic solver would be faced, including: resolution, topology, geometry, boundary conditions, domain discretization regularity, dimensionality, etc. In this work, we build a solver, satisfying these properties, where all the components are based on neural message passing, replacing all heuristically designed components in the computation graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. In order to encourage stability in training autoregressive models, we put forward a method that is based on the principle of zero-stability, posing stability as a domain adaptation problem. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
Cooperative Medianet Innovation Center（协同媒体网络创新中心）；Shanghai Jiao Tong University（上海交通大学）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）
Visual relationship detection aims to detect the interactions between objects in an image; however, this task suffers from combinatorial explosion due to the variety of objects and interactions. Since the interactions associated with the same object are dependent, we explore the dependency of interactions to reduce the search space. We explicitly model objects and interactions by an interaction graph and then propose a message-passing-style algorithm to propagate the contextual information. We thus call the proposed method neural message passing (NMP). We further integrate language priors and …
We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale sequence models and graph gating modules in the encoder and processor, respectively. Benchmark numerical experiments are presented to demonstrate that the proposed algorithm outperforms baselines, particularly on a PDE with a range of spatial and temporal scales.
We propose a new family of message passing techniques for MAP estimation in graphical models which we call {\em Sequential Reweighted Message Passing} (SRMP). Special cases include well-known techniques such as {\em Min-Sum Diffusion} (MSD) and a faster {\em Sequential Tree-Reweighted Message Passing} (TRW-S). Importantly, our derivation is simpler than the original derivation of TRW-S, and does not involve a decomposition into trees. This allows easy generalizations. We present such a generalization for the case of higher-order graphical models, and test it on several real-world problems with…
Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学研究所）；UCLA（加州大学洛杉矶分校）；Center for Scalable Data Analytics and Artificial Intelligence（可扩展数据与人工智能中心）；University of Oxford（牛津大学）
One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE solvers through learning using message-passing models. We first introduce domain invariant features for PDE-data inspired by classical PDE solvers for an efficient physical representation. Next, we use graphs to represent PDE-data on an unstructured mesh and show that message passing graph neural networks (MPGNN) can parameterize governing equations, and as a …
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Shanghai Jiao Tong University（上海交通大学）
Graph neural networks (GNNs) have achieved champion in wide applications. Neural message passing is a typical key module for feature propagation by aggregating neighboring features. In this work, we propose a new message passing based on multiscale framelet transforms, called Framelet Message Passing. Different from traditional spatial methods, it integrates framelet representation of neighbor nodes from multiple hops away in node message update. We also propose a continuous message passing using neural ODE solvers. It turns both discrete and continuous cases can provably achieve network stabi…
The paper proposes a new message passing algorithm for cycle-free factor graphs. The proposed "entropy message passing" (EMP) algorithm may be viewed as sum-product message passing over the entropy semiring, which has previously appeared in automata theory. The primary use of EMP is to compute the entropy of a model. However, EMP can also be used to compute expressions that appear in expectation maximization and in gradient descent algorithms.
Washington University in St. Louis（圣路易斯华盛顿大学）；Washington University School of Medicine（华盛顿大学医学院）；Peking University（北京大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
The most popular design paradigm for Graph Neural Networks (GNNs) is 1-hop message passing -- aggregating information from 1-hop neighbors repeatedly. However, the expressive power of 1-hop message passing is bounded by the Weisfeiler-Lehman (1-WL) test. Recently, researchers extended 1-hop message passing to K-hop message passing by aggregating information from K-hop neighbors of nodes simultaneously. However, there is no work on analyzing the expressive power of K-hop message passing. In this work, we theoretically characterize the expressive power of K-hop message passing. Specifically, we …
MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；Microsoft Research（微软研究院）
Generative models provide a powerful framework for probabilistic reasoning. However, in many domains their use has been hampered by the practical difficulties of inference. This is particularly the case in computer vision, where models of the imaging process tend to be large, loopy and layered. For this reason bottom-up conditional models have traditionally dominated in such domains. We find that widely-used, general-purpose message passing inference algorithms such as Expectation Propagation (EP) and Variational Message Passing (VMP) fail on the simplest of vision models. With these models in…
The message passing framework is the foundation of the immense success enjoyed by graph neural networks (GNNs) in recent years. In spite of its elegance, there exist many problems it provably cannot solve over given input graphs. This has led to a surge of research on going "beyond message passing", building GNNs which do not suffer from those limitations -- a term which has become ubiquitous in regular discourse. However, have those methods truly moved beyond message passing? In this position paper, I argue about the dangers of using this term -- especially when teaching graph representation …
Link Prediction with Untrained Message Passing Layers
Chair of Machine Learning for Complex Networks（复杂网络机器学习教授席位）；Center for Artificial Intelligence and Data Science (CAIDAS)（人工智能与数据科学中心）；Julius-Maximilians-Universität Würzburg（维尔堡大学）
Message passing is a fundamental procedure for graph neural networks in the field of graph representation learning. Based on the homophily assumption, the current message passing always aggregates features of connected nodes, such as the graph Laplacian smoothing process. However, real-world graphs tend to be noisy and/or non-smooth. The homophily assumption does not always hold, leading to sub-optimal results. A revised message passing method needs to maintain each node's discriminative ability when aggregating the message from neighbors. To this end, we propose a Memory-based Message Passing…
Based on prior work by Eckford, it is shown how expectation maximization (EM) may be viewed, and used, as a message passing algorithm in factor graphs.
In recent studies, neural message passing has proved to be an effective way to design graph neural networks (GNNs), which have achieved state-of-the-art performance in many graph-based tasks. However, current neural-message passing architectures typically need to perform an expensive recursive neighborhood expansion in multiple rounds and consequently suffer from a scalability issue. Moreover, most existing neural-message passing schemes are inflexible since they are restricted to fixed-hop neighborhoods and insensitive to the actual demands of different nodes. We circumvent these limitations …
A general graph-structured neural network architecture operates on graphs through two core components: (1) complex enough message functions; (2) a fixed information aggregation process. In this paper, we present the Policy Message Passing algorithm, which takes a probabilistic perspective and reformulates the whole information aggregation as stochastic sequential processes. The algorithm works on a much larger search space, utilizes reasoning history to perform inference, and is robust to noisy edges. We apply our algorithm to multiple complex graph reasoning and prediction tasks and show that…
Message passing on a factor graph is a powerful paradigm for the coding of approximate inference algorithms for arbitrarily graphical large models. The notion of a factor graph fragment allows for compartmentalization of algebra and computer code. We show that the Inverse G-Wishart family of distributions enables fundamental variational message passing factor graph fragments to be expressed elegantly and succinctly. Such fragments arise in models for which approximate inference concerning covariance matrix or variance parameters is made, and are ubiquitous in contemporary statistics and machin…
University of Chinese Academy of Sciences（中国科学院大学）；University of Science and Technology of China（中国科学技术大学）；Peng Cheng Laboratory（鹏城实验室）
Graph neural networks (GNNs) emerged recently as a standard toolkit for learning from data on graphs. Current GNN designing works depend on immense human expertise to explore different message-passing mechanisms, and require manual enumeration to determine the proper message-passing depth. Inspired by the strong searching capability of neural architecture search (NAS) in CNN, this paper proposes Graph Neural Architecture Search (GNAS) with novel-designed search space. The GNAS can automatically learn better architecture with the optimal depth of message passing on the graph. Specifically, we d…
The mechanism of message passing in graph neural networks (GNNs) is still mysterious. Apart from convolutional neural networks, no theoretical origin for GNNs has been proposed. To our surprise, message passing can be best understood in terms of power iteration. By fully or partly removing activation functions and layer weights of GNNs, we propose subspace power iteration clustering (SPIC) models that iteratively learn with only one aggregator. Experiments show that our models extend GNNs and enhance their capability to process random featured networks. Moreover, we demonstrate the redundancy …
Eindhoven University of Technology（埃因霍温理工大学）；GN Hearing（GN Hearing 公司）
We introduce Reactive Message Passing (RMP) as a framework for executing schedule-free, robust and scalable message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style that only describes how nodes in a factor graph react to changes in connected nodes. The absence of a fixed message passing schedule improves robustness, scalability and execution time of the inference procedure. We also present ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-de…
Institute of Computing Technology, CAS（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Agriculture Information Institute, CAAS（中国农业科学院农业信息研究所）；Tsinghua University（清华大学）；Peng Cheng Laboratory（鹏城实验室）
In recent years, contrastive learning achieves impressive results on self-supervised visual representation learning, but there still lacks a rigorous understanding of its learning dynamics. In this paper, we show that if we cast a contrastive objective equivalently into the feature space, then its learning dynamics admits an interpretable form. Specifically, we show that its gradient descent corresponds to a specific message passing scheme on the corresponding augmentation graph. Based on this perspective, we theoretically characterize how contrastive learning gradually learns discriminative f…
Communicating and sharing intelligence among agents is an important facet of achieving Artificial General Intelligence. As a first step towards this challenge, we introduce a novel framework for image generation: Message Passing Multi-Agent Generative Adversarial Networks (MPM GANs). While GANs have recently been shown to be very effective for image generation and other tasks, these networks have been limited to mostly single generator-discriminator networks. We show that we can obtain multi-agent GANs that communicate through message passing to achieve better image generation. The objectives …
We present a novel method for learning the weights of an artificial neural network - a Message Passing Learning Protocol (MPLP). In MPLP, we abstract every operations occurring in ANNs as independent agents. Each agent is responsible for ingesting incoming multidimensional messages from other agents, updating its internal state, and generating multidimensional messages to be passed on to neighbouring agents. We demonstrate the viability of MPLP as opposed to traditional gradient-based approaches on simple feed-forward neural networks, and present a framework capable of generalizing to non-trad…
Seoul National University（首尔大学）；University of Birmingham（伯明翰大学）
Most of the existing literature regarding hyperbolic embedding concentrate upon supervised learning, whereas the use of unsupervised hyperbolic embedding is less well explored. In this paper, we analyze how unsupervised tasks can benefit from learned representations in hyperbolic space. To explore how well the hierarchical structure of unlabeled data can be represented in hyperbolic spaces, we design a novel hyperbolic message passing auto-encoder whose overall auto-encoding is performed in hyperbolic space. The proposed model conducts auto-encoding the networks via fully utilizing hyperbolic …
Knowledge graph completion aims to predict missing relations between entities in a knowledge graph. In this work, we propose a relational message passing method for knowledge graph completion. Different from existing embedding-based methods, relational message passing only considers edge features (i.e., relation types) without entity IDs in the knowledge graph, and passes relational messages among edges iteratively to aggregate neighborhood information. Specifically, two kinds of neighborhood topology are modeled for a given entity pair under the relational message passing framework: (1) Relat…
University College London（伦敦大学学院）；Google Deepmind（谷歌DeepMind）
We propose to learn a kernel-based message operator which takes as input all expectation propagation (EP) incoming messages to a factor node and produces an outgoing message. In ordinary EP, computing an outgoing message involves estimating a multivariate integral which may not have an analytic expression. Learning such an operator allows one to bypass the expensive computation of the integral during inference by directly mapping all incoming messages into an outgoing message. The operator can be learned from training data (examples of input and output messages) which allows automated inferenc…
Technische Universität Berlin（柏林工业大学）；University of Birmingham（伯明翰大学）
We analyze the random sequential dynamics of a message passing algorithm for Ising models with random interactions in the large system limit. We derive exact results for the two-time correlation functions and the speed of convergence. The {\em de Almedia-Thouless} stability criterion of the static problem is found to be necessary and sufficient for the global convergence of the random sequential dynamics.
We present a hierarchical neural message passing architecture for learning on molecular graphs. Our model takes in two complementary graph representations: the raw molecular graph representation and its associated junction tree, where nodes represent meaningful clusters in the original graph, e.g., rings or bridged compounds. We then proceed to learn a molecule's representation by passing messages inside each graph, and exchange messages between the two representations using a coarse-to-fine and fine-to-coarse information flow. Our method is able to overcome some of the restrictions known from…
the Hong Kong University of Science and Technology（香港科技大学）
Graph neural networks (GNNs) and message passing neural networks (MPNNs) have been proven to be expressive for subgraph structures in many applications. Some applications in heterogeneous graphs require explicit edge modeling, such as subgraph isomorphism counting and matching. However, existing message passing mechanisms are not designed well in theory. In this paper, we start from a particular edge-to-vertex transform and exploit the isomorphism property in the edge-to-vertex dual graphs. We prove that searching isomorphisms on the original graph is equivalent to searching on its dual graph.…
Shanghai Jiao Tong University（上海交通大学）；Chinese Academy of Sciences（中国科学院）
Most graph neural networks follow the message passing mechanism. However, it faces the over-smoothing problem when multiple times of message passing is applied to a graph, causing indistinguishable node representations and prevents the model to effectively learn dependencies between farther-away nodes. On the other hand, features of neighboring nodes with different labels are likely to be falsely mixed, resulting in the heterophily problem. In this work, we propose to order the messages passing into the node representation, with specific blocks of neurons targeted for message passing within sp…
We address the problem of computing approximate marginals in Gaussian probabilistic models by using mean field and fractional Bethe approximations. We define the Gaussian fractional Bethe free energy in terms of the moment parameters of the approximate marginals, derive a lower and an upper bound on the fractional Bethe free energy and establish a necessary condition for the lower bound to be bounded from below. It turns out that the condition is identical to the pairwise normalizability condition, which is known to be a sufficient condition for the convergence of the message passing algorithm…
Stanford University（斯坦福大学）；University of California, Berkeley（加州大学伯克利分校）
Maximum a posteriori (MAP) inference in discrete-valued Markov random fields is a fundamental problem in machine learning that involves identifying the most likely configuration of random variables given a distribution. Due to the difficulty of this combinatorial problem, linear programming (LP) relaxations are commonly used to derive specialized message passing algorithms that are often interpreted as coordinate descent on the dual LP. To achieve more desirable computational properties, a number of methods regularize the LP with an entropy term, leading to a class of smooth message passing al…
Artificial Intelligence Technology Engineering Research Center, Open University of Henan（河南开放大学人工智能技术工程研究中心）；University of Wollongong（伍伦贡大学）；Weizmann Institute of Science（魏茨曼科学研究所）；University of Sydney（悉尼大学）
We consider matrix factorization (MF) with certain constraints, which finds wide applications in various areas. Leveraging variational inference (VI) and unitary approximate message passing (UAMP), we develop a Bayesian approach to MF with an efficient message passing implementation, called UAMPMF. With proper priors imposed on the factor matrices, UAMPMF can be used to solve many problems that can be formulated as MF, such as non negative matrix factorization, dictionary learning, compressive sensing with matrix uncertainty, robust principal component analysis, and sparse matrix factorization…
University of Electronic Science and Technology of China（电子科技大学）
Nowadays, Graph Neural Networks (GNNs) following the Message Passing paradigm become the dominant way to learn on graphic data. Models in this paradigm have to spend extra space to look up adjacent nodes with adjacency matrices and extra time to aggregate multiple messages from adjacent nodes. To address this issue, we develop a method called LinkDist that distils self-knowledge from connected node pairs into a Multi-Layer Perceptron (MLP) without the need to aggregate messages. Experiment with 8 real-world datasets shows the MLP derived from LinkDist can predict the label of a node without kn…
Message-passing algorithms have emerged as powerful techniques for approximate inference in graphical models. When these algorithms converge, they can be shown to find local (or sometimes even global) optima of variational formulations to the inference problem. But many of the most popular algorithms are not guaranteed to converge. This has lead to recent interest in convergent message-passing algorithms. In this paper, we present a unified view of convergent message-passing algorithms. We present a simple derivation of an abstract algorithm, tree-consistency bound optimization (TCBO) that is …
Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes). They do not, however, consider the spatial direction from one atom to another, despite directional information playing a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a …
Graph Neural Networks (GNNs) are the subject of intense focus by the machine learning community for problems involving relational reasoning. GNNs can be broadly divided into spatial and spectral approaches. Spatial approaches use a form of learned message-passing, in which interactions among vertices are computed locally, and information propagates over longer distances on the graph with greater numbers of message-passing steps. Spectral approaches use eigendecompositions of the graph Laplacian to produce a generalization of spatial convolutions to graph structured data which access informatio…
Message-passing has proved to be an effective way to design graph neural networks, as it is able to leverage both permutation equivariance and an inductive bias towards learning local structures in order to achieve good generalization. However, current message-passing architectures have a limited representation power and fail to learn basic topological properties of graphs. We address this problem and propose a powerful and equivariant message-passing framework based on two ideas: first, we propagate a one-hot encoding of the nodes, in addition to the features, in order to learn a local contex…
Macroeconomic Message Passing for Anticipating Foreign Exchange Regime Changes: A Deep Logical Learning Approach using Graph Tsetlin Machines
Centre for Artificial Intelligence Research, University of Agder（人工智能研究中心，阿格德大学）；La Jolla Private Wealth Group, Wells Fargo Advisors（拉贾尔私人财富集团，沃尔索尔顾问公司）
We show that a message-passing process allows to store in binary "material" synapses a number of random patterns which almost saturates the information theoretic bounds. We apply the learning algorithm to networks characterized by a wide range of different connection topologies and of size comparable with that of biological systems (e.g. $n\simeq10^{5}-10^{6}$). The algorithm can be turned into an on-line --fault tolerant-- learning protocol of potential interest in modeling aspects of synaptic plasticity and in building neuromorphic devices.
