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Autoencoders are the simplest neural network for unsupervised learning, and thus an ideal framework for studying feature learning. While a detailed understanding of the dynamics of linear autoencoders has recently been obtained, the study of non-linear autoencoders has been hindered by the technical difficulty of handling training data with non-trivial correlations - a fundamental prerequisite for feature extraction. Here, we study the dynamics of feature learning in non-linear, shallow autoencoders. We derive a set of asymptotically exact equations that describe the generalisation dynamics of autoencoders trained with stochastic gradient descent (SGD) in the limit of high-dimensional inputs. These equations reveal that autoencoders learn the leading principal components of their inputs sequentially. An analysis of the long-time dynamics explains the failure of sigmoidal autoencoders to learn with tied weights, and highlights the importance of training the bias in ReLU autoencoders. Building on previous results for linear networks, we analyse a modification of the vanilla SGD algorithm which allows learning of the exact principal components. Finally, we show that our equations accurately describe the generalisation dynamics of non-linear autoencoders on realistic datasets such as CIFAR10.

A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics is proposed and validated using high-temperature auto-ignitions, perfectly stirred reactors (PSR), and one-dimensional freely propagating flames of n-heptane/air mixtures. The mechanism reduction is modeled as an optimization problem on Boolean space, where a Boolean vector, each entry corresponding to a species, represents a reduced mechanism. The optimization goal is to minimize the reduced mechanism size given the error tolerance of a group of pre-selected benchmark quantities. The key idea of the DeePMR is to employ a deep neural network (DNN) to formulate the objective function in the optimization problem. In order to explore high dimensional Boolean space efficiently, an iterative DNN-assisted data sampling and DNN training procedure are implemented. The results show that DNN-assistance improves sampling efficiency significantly, selecting only $10^5$ samples out of $10^{34}$ possible samples for DNN to achieve sufficient accuracy. The results demonstrate the capability of the DNN to recognize key species and reasonably predict reduced mechanism performance. The well-trained DNN guarantees the optimal reduced mechanism by solving an inverse optimization problem. By comparing ignition delay times, laminar flame speeds, temperatures in PSRs, the resulting skeletal mechanism has fewer species (45 species) but the same level of accuracy as the skeletal mechanism (56 species) obtained by the Path Flux Analysis (PFA) method. In addition, the skeletal mechanism can be further reduced to 28 species if only considering atmospheric, near-stoichiometric conditions (equivalence ratio between 0.6 and 1.2). The DeePMR provides an innovative way to perform model reduction and demonstrates the great potential of data-driven methods in the combustion area.

We propose a first-order autoregressive (i.e. AR(1)) model for dynamic network processes in which edges change over time while nodes remain unchanged. The model depicts the dynamic changes explicitly. It also facilitates simple and efficient statistical inference methods including a permutation test for diagnostic checking for the fitted network models. The proposed model can be applied to the network processes with various underlying structures but with independent edges. As an illustration, an AR(1) stochastic block model has been investigated in depth, which characterizes the latent communities by the transition probabilities over time. This leads to a new and more effective spectral clustering algorithm for identifying the latent communities. We have derived a finite sample condition under which the perfect recovery of the community structure can be achieved by the newly defined spectral clustering algorithm. Furthermore the inference for a change point is incorporated into the AR(1) stochastic block model to cater for possible structure changes. We have derived the explicit error rates for the maximum likelihood estimator of the change-point. Application with three real data sets illustrates both relevance and usefulness of the proposed AR(1) models and the associate inference methods.

We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks, for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse boundary value problems. Traditional neural networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with the linear operators that correspond to the data in such inverse problems. We therefore introduce a new family that resembles a standard neural network architecture, but where the input data acts multiplicatively on vectors. Motivated by compact operators appearing in boundary control and the analysis of inverse boundary value problems for the wave equation, we promote structure and sparsity in selected weight matrices in the network. After describing this architecture, we study its representation properties as well as its approximation properties. We furthermore show that an explicit regularization can be introduced that can be derived from the mathematical analysis of the mentioned inverse problems, and which leads to certain guarantees on the generalization properties. We observe that the sparsity of the weight matrices improves the generalization estimates. Lastly, we discuss how operator recurrent networks can be viewed as a deep learning analogue to deterministic algorithms such as boundary control for reconstructing the unknown wavespeed in the acoustic wave equation from boundary measurements.

Molecular dynamics simulations are a cornerstone in science, allowing to investigate from the system's thermodynamics to analyse intricate molecular interactions. In general, to create extended molecular trajectories can be a computationally expensive process, for example, when running $ab-initio$ simulations. Hence, repeating such calculations to either obtain more accurate thermodynamics or to get a higher resolution in the dynamics generated by a fine-grained quantum interaction can be time- and computationally-consuming. In this work, we explore different machine learning (ML) methodologies to increase the resolution of molecular dynamics trajectories on-demand within a post-processing step. As a proof of concept, we analyse the performance of bi-directional neural networks such as neural ODEs, Hamiltonian networks, recurrent neural networks and LSTMs, as well as the uni-directional variants as a reference, for molecular dynamics simulations (here: the MD17 dataset). We have found that Bi-LSTMs are the best performing models; by utilizing the local time-symmetry of thermostated trajectories they can even learn long-range correlations and display high robustness to noisy dynamics across molecular complexity. Our models can reach accuracies of up to 10$^{-4}$ angstroms in trajectory interpolation, while faithfully reconstructing several full cycles of unseen intricate high-frequency molecular vibrations, rendering the comparison between the learned and reference trajectories indistinguishable. The results reported in this work can serve (1) as a baseline for larger systems, as well as (2) for the construction of better MD integrators.

Understanding the inner workings of deep neural networks (DNNs) is essential to provide trustworthy artificial intelligence techniques for practical applications. Existing studies typically involve linking semantic concepts to units or layers of DNNs, but fail to explain the inference process. In this paper, we introduce neural architecture disentanglement (NAD) to fill the gap. Specifically, NAD learns to disentangle a pre-trained DNN into sub-architectures according to independent tasks, forming information flows that describe the inference processes. We investigate whether, where, and how the disentanglement occurs through experiments conducted with handcrafted and automatically-searched network architectures, on both object-based and scene-based datasets. Based on the experimental results, we present three new findings that provide fresh insights into the inner logic of DNNs. First, DNNs can be divided into sub-architectures for independent tasks. Second, deeper layers do not always correspond to higher semantics. Third, the connection type in a DNN affects how the information flows across layers, leading to different disentanglement behaviors. With NAD, we further explain why DNNs sometimes give wrong predictions. Experimental results show that misclassified images have a high probability of being assigned to task sub-architectures similar to the correct ones. Code will be available at: //github.com/hujiecpp/NAD.

Deep learning methods for graphs achieve remarkable performance on many node-level and graph-level prediction tasks. However, despite the proliferation of the methods and their success, prevailing Graph Neural Networks (GNNs) neglect subgraphs, rendering subgraph prediction tasks challenging to tackle in many impactful applications. Further, subgraph prediction tasks present several unique challenges, because subgraphs can have non-trivial internal topology, but also carry a notion of position and external connectivity information relative to the underlying graph in which they exist. Here, we introduce SUB-GNN, a subgraph neural network to learn disentangled subgraph representations. In particular, we propose a novel subgraph routing mechanism that propagates neural messages between the subgraph's components and randomly sampled anchor patches from the underlying graph, yielding highly accurate subgraph representations. SUB-GNN specifies three channels, each designed to capture a distinct aspect of subgraph structure, and we provide empirical evidence that the channels encode their intended properties. We design a series of new synthetic and real-world subgraph datasets. Empirical results for subgraph classification on eight datasets show that SUB-GNN achieves considerable performance gains, outperforming strong baseline methods, including node-level and graph-level GNNs, by 12.4% over the strongest baseline. SUB-GNN performs exceptionally well on challenging biomedical datasets when subgraphs have complex topology and even comprise multiple disconnected components.

It is not until recently that graph neural networks (GNNs) are adopted to perform graph representation learning, among which, those based on the aggregation of features within the neighborhood of a node achieved great success. However, despite such achievements, GNNs illustrate defects in identifying some common structural patterns which, unfortunately, play significant roles in various network phenomena. In this paper, we propose GraLSP, a GNN framework which explicitly incorporates local structural patterns into the neighborhood aggregation through random anonymous walks. Specifically, we capture local graph structures via random anonymous walks, powerful and flexible tools that represent structural patterns. The walks are then fed into the feature aggregation, where we design various mechanisms to address the impact of structural features, including adaptive receptive radius, attention and amplification. In addition, we design objectives that capture similarities between structures and are optimized jointly with node proximity objectives. With the adequate leverage of structural patterns, our model is able to outperform competitive counterparts in various prediction tasks in multiple datasets.

Matter evolved under influence of gravity from minuscule density fluctuations. Non-perturbative structure formed hierarchically over all scales, and developed non-Gaussian features in the Universe, known as the Cosmic Web. To fully understand the structure formation of the Universe is one of the holy grails of modern astrophysics. Astrophysicists survey large volumes of the Universe and employ a large ensemble of computer simulations to compare with the observed data in order to extract the full information of our own Universe. However, to evolve trillions of galaxies over billions of years even with the simplest physics is a daunting task. We build a deep neural network, the Deep Density Displacement Model (hereafter D$^3$M), to predict the non-linear structure formation of the Universe from simple linear perturbation theory. Our extensive analysis, demonstrates that D$^3$M outperforms the second order perturbation theory (hereafter 2LPT), the commonly used fast approximate simulation method, in point-wise comparison, 2-point correlation, and 3-point correlation. We also show that D$^3$M is able to accurately extrapolate far beyond its training data, and predict structure formation for significantly different cosmological parameters. Our study proves, for the first time, that deep learning is a practical and accurate alternative to approximate simulations of the gravitational structure formation of the Universe.

Modern communication networks have become very complicated and highly dynamic, which makes them hard to model, predict and control. In this paper, we develop a novel experience-driven approach that can learn to well control a communication network from its own experience rather than an accurate mathematical model, just as a human learns a new skill (such as driving, swimming, etc). Specifically, we, for the first time, propose to leverage emerging Deep Reinforcement Learning (DRL) for enabling model-free control in communication networks; and present a novel and highly effective DRL-based control framework, DRL-TE, for a fundamental networking problem: Traffic Engineering (TE). The proposed framework maximizes a widely-used utility function by jointly learning network environment and its dynamics, and making decisions under the guidance of powerful Deep Neural Networks (DNNs). We propose two new techniques, TE-aware exploration and actor-critic-based prioritized experience replay, to optimize the general DRL framework particularly for TE. To validate and evaluate the proposed framework, we implemented it in ns-3, and tested it comprehensively with both representative and randomly generated network topologies. Extensive packet-level simulation results show that 1) compared to several widely-used baseline methods, DRL-TE significantly reduces end-to-end delay and consistently improves the network utility, while offering better or comparable throughput; 2) DRL-TE is robust to network changes; and 3) DRL-TE consistently outperforms a state-ofthe-art DRL method (for continuous control), Deep Deterministic Policy Gradient (DDPG), which, however, does not offer satisfying performance.