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This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks. In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nystrom-based interpolation method.

This paper introduces methods and a novel toolbox that efficiently integrates any high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. Semi-analytical integration of the RDE is done with the Local Linearization scheme for each neural mass model, which is the only scheme guaranteeing dynamical fidelity to the original continuous-time nonlinear dynamic. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism, as shown by the Moore-Penrose diagram of the algorithm. This ease of implementation includes models with distributed-delay CTs. We achieve high computational efficiency by using a tensor representation of the model that leverages semi-analytic expressions to integrate the Random Differential Equations (RDEs) underlying the NMM. We discretized the state equation with Local Linearization via an algebraic formulation. This approach increases numerical integration speed and efficiency, a crucial aspect of large-scale NMM simulations. To illustrate the usefulness of the toolbox, we simulate both a single Zetterberg-Jansen-Rit (ZJR) cortical column and an interconnected population of such columns. These examples illustrate the consequence of modifying the CT in these models, especially by introducing distributed delays. We provide an open-source Matlab live script for the toolbox.

Classifiers learnt from data are increasingly being used as components in systems where safety is a critical concern. In this work, we present a formal notion of safety for classifiers via constraints called safe-ordering constraints. These constraints relate requirements on the order of the classes output by a classifier to conditions on its input, and are expressive enough to encode various interesting examples of classifier safety specifications from the literature. For classifiers implemented using neural networks, we also present a run-time mechanism for the enforcement of safe-ordering constraints. Our approach is based on a self-correcting layer, which provably yields safe outputs regardless of the characteristics of the classifier input. We compose this layer with an existing neural network classifier to construct a self-correcting network (SC-Net), and show that in addition to providing safe outputs, the SC-Net is guaranteed to preserve the classification accuracy of the original network whenever possible. Our approach is independent of the size and architecture of the neural network used for classification, depending only on the specified property and the dimension of the network's output; thus it is scalable to large state-of-the-art networks. We show that our approach can be optimized for a GPU, introducing run-time overhead of less than 1ms on current hardware -- even on large, widely-used networks containing hundreds of thousands of neurons and millions of parameters.

Information about action costs is critical for real-world AI planning applications. Rather than rely solely on declarative action models, recent approaches also use black-box external action cost estimators, often learned from data, that are applied during the planning phase. These, however, can be computationally expensive, and produce uncertain values. In this paper we suggest a generalization of deterministic planning with action costs that allows selecting between multiple estimators for action cost, to balance computation time against bounded estimation uncertainty. This enables a much richer -- and correspondingly more realistic -- problem representation. Importantly, it allows planners to bound plan accuracy, thereby increasing reliability, while reducing unnecessary computational burden, which is critical for scaling to large problems. We introduce a search algorithm, generalizing $A^*$, that solves such planning problems, and additional algorithmic extensions. In addition to theoretical guarantees, extensive experiments show considerable savings in runtime compared to alternatives.

Nowadays, data are richly accessible to accumulate, and the increasingly powerful capability with computing offers reasonable ease of handling big data. This remarkable scenario leads to a new way for solving some control problems which was previously hard to analyze and solve. In this paper, a new type of control methods, namely control with patterns (CWP), is proposed to handle data sets corresponding to nonlinear dynamical systems subject to a discrete control constraint set. For data sets of this kind, a new definition, namely exponential attraction on data sets, is proposed to describe nonlinear dynamical systems under consideration. Based on the data sets and parameterized Lyapunov functions, the problem for exponential attraction on data sets is converted to a pattern classification one. Furthermore, the controller design is proposed accordingly, where the pattern classification function is used to decide which control element in the control set should be employed. Illustrative examples are given to show the effectiveness of the proposed CWP.

Many future technologies rely on neural networks, but verifying the correctness of their behavior remains a major challenge. It is known that neural networks can be fragile in the presence of even small input perturbations, yielding unpredictable outputs. The verification of neural networks is therefore vital to their adoption, and a number of approaches have been proposed in recent years. In this paper we focus on semidefinite programming (SDP) based techniques for neural network verification, which are particularly attractive because they can encode expressive behaviors while ensuring a polynomial time decision. Our starting point is the DeepSDP framework proposed by Fazlyab et al, which uses quadratic constraints to abstract the verification problem into a large-scale SDP. When the size of the neural network grows, however, solving this SDP quickly becomes intractable. Our key observation is that by leveraging chordal sparsity and specific parametrizations of DeepSDP, we can decompose the primary computational bottleneck of DeepSDP -- a large linear matrix inequality (LMI) -- into an equivalent collection of smaller LMIs. Our parametrization admits a tunable parameter, allowing us to trade-off efficiency and accuracy in the verification procedure. We call our formulation Chordal-DeepSDP, and provide experimental evaluation to show that it can: (1) effectively increase accuracy with the tunable parameter and (2) outperform DeepSDP on deeper networks.

As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.

Dynamic neural network is an emerging research topic in deep learning. Compared to static models which have fixed computational graphs and parameters at the inference stage, dynamic networks can adapt their structures or parameters to different inputs, leading to notable advantages in terms of accuracy, computational efficiency, adaptiveness, etc. In this survey, we comprehensively review this rapidly developing area by dividing dynamic networks into three main categories: 1) instance-wise dynamic models that process each instance with data-dependent architectures or parameters; 2) spatial-wise dynamic networks that conduct adaptive computation with respect to different spatial locations of image data and 3) temporal-wise dynamic models that perform adaptive inference along the temporal dimension for sequential data such as videos and texts. The important research problems of dynamic networks, e.g., architecture design, decision making scheme, optimization technique and applications, are reviewed systematically. Finally, we discuss the open problems in this field together with interesting future research directions.

Traditional methods for link prediction can be categorized into three main types: graph structure feature-based, latent feature-based, and explicit feature-based. Graph structure feature methods leverage some handcrafted node proximity scores, e.g., common neighbors, to estimate the likelihood of links. Latent feature methods rely on factorizing networks' matrix representations to learn an embedding for each node. Explicit feature methods train a machine learning model on two nodes' explicit attributes. Each of the three types of methods has its unique merits. In this paper, we propose SEAL (learning from Subgraphs, Embeddings, and Attributes for Link prediction), a new framework for link prediction which combines the power of all the three types into a single graph neural network (GNN). GNN is a new type of neural network which directly accepts graphs as input and outputs their labels. In SEAL, the input to the GNN is a local subgraph around each target link. We prove theoretically that our local subgraphs also reserve a great deal of high-order graph structure features related to link existence. Another key feature is that our GNN can naturally incorporate latent features and explicit features. It is achieved by concatenating node embeddings (latent features) and node attributes (explicit features) in the node information matrix for each subgraph, thus combining the three types of features to enhance GNN learning. Through extensive experiments, SEAL shows unprecedentedly strong performance against a wide range of baseline methods, including various link prediction heuristics and network embedding methods.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.