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In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs and variants -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving irregularly in time and observed at arbitrary spatial resolutions. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can parameterize solution operators of SPDEs depending simultaneously on the initial condition and a realization of the driving noise. By performing operations in the spectral domain, we show how a Neural SPDE can be evaluated in two ways, either by calling an ODE solver (emulating a spectral Galerkin scheme), or by solving a fixed point problem. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers.

Video action recognition has been partially addressed by the CNNs stacking of fixed-size 3D kernels. However, these methods may under-perform for only capturing rigid spatial-temporal patterns in single-scale spaces, while neglecting the scale variances across different action primitives. To overcome this limitation, we propose to learn the optimal-scale kernels from the data. More specifically, an \textit{action perceptron synthesizer} is proposed to generate the kernels from a bag of fixed-size kernels that are interacted by dense routing paths. To guarantee the interaction richness and the information capacity of the paths, we design the novel \textit{optimized feature fusion layer}. This layer establishes a principled universal paradigm that suffices to cover most of the current feature fusion techniques (e.g., channel shuffling, and channel dropout) for the first time. By inserting the \textit{synthesizer}, our method can easily adapt the traditional 2D CNNs to the video understanding tasks such as action recognition with marginal additional computation cost. The proposed method is thoroughly evaluated over several challenging datasets (i.e., Somehting-to-Somthing, Kinetics and Diving48) that highly require temporal reasoning or appearance discriminating, achieving new state-of-the-art results. Particularly, our low-resolution model outperforms the recent strong baseline methods, i.e., TSM and GST, with less than 30\% of their computation cost.

In this paper, we propose a novel design, called MixNN, for protecting deep learning model structure and parameters. The layers in a deep learning model of MixNN are fully decentralized. It hides communication address, layer parameters and operations, and forward as well as backward message flows among non-adjacent layers using the ideas from mix networks. MixNN has following advantages: 1) an adversary cannot fully control all layers of a model including the structure and parameters, 2) even some layers may collude but they cannot tamper with other honest layers, 3) model privacy is preserved in the training phase. We provide detailed descriptions for deployment. In one classification experiment, we compared a neural network deployed in a virtual machine with the same one using the MixNN design on the AWS EC2. The result shows that our MixNN retains less than 0.001 difference in terms of classification accuracy, while the whole running time of MixNN is about 7.5 times slower than the one running on a single virtual machine.

Recently, the enactment of privacy regulations has promoted the rise of the machine unlearning paradigm. Existing studies of machine unlearning mainly focus on sample-wise unlearning, such that a learnt model will not expose user's privacy at the sample level. Yet we argue that such ability of selective removal should also be presented at the attribute level, especially for the attributes irrelevant to the main task, e.g., whether a person recognized in a face recognition system wears glasses or the age range of that person. Through a comprehensive literature review, it is found that existing studies on attribute-related problems like fairness and de-biasing learning cannot address the above concerns properly. To bridge this gap, we propose a paradigm of selectively removing input attributes from feature representations which we name `attribute unlearning'. In this paradigm, certain attributes will be accurately captured and detached from the learned feature representations at the stage of training, according to their mutual information. The particular attributes will be progressively eliminated along with the training procedure towards convergence, while the rest of attributes related to the main task are preserved for achieving competitive model performance. Considering the computational complexity during the training process, we not only give a theoretically approximate training method, but also propose an acceleration scheme to speed up the training process. We validate our method by spanning several datasets and models and demonstrate that our design can preserve model fidelity and reach prevailing unlearning efficacy with high efficiency. The proposed unlearning paradigm builds a foundation for future machine unlearning system and will become an essential component of the latest privacy-related legislation.

We design the helicity-conservative physics-informed neural network model for the Navier-Stokes equation in the ideal case. The key is to provide an appropriate PDE model as loss function so that its neural network solutions produce helicity conservation. Physics-informed neural network model is based on the strong form of PDE. We show that the relevant helicity-conservative finite element method based on the weak formulation of PDE can be somewhat different. More precisely, we compares the PINN formulation and the finite element method based on the weak formulation for conserving helicity and argues that for the conservation, strong PDE is more natural. Our result is justified by theory as well. Furthermore, a couple of numerical calculations are demonstrated to confirm our theoretical finding.

Synthesis of ergodic, stationary visual patterns is widely applicable in texturing, shape modeling, and digital content creation. The wide applicability of this technique thus requires the pattern synthesis approaches to be scalable, diverse, and authentic. In this paper, we propose an exemplar-based visual pattern synthesis framework that aims to model the inner statistics of visual patterns and generate new, versatile patterns that meet the aforementioned requirements. To this end, we propose an implicit network based on generative adversarial network (GAN) and periodic encoding, thus calling our network the Implicit Periodic Field Network (IPFN). The design of IPFN ensures scalability: the implicit formulation directly maps the input coordinates to features, which enables synthesis of arbitrary size and is computationally efficient for 3D shape synthesis. Learning with a periodic encoding scheme encourages diversity: the network is constrained to model the inner statistics of the exemplar based on spatial latent codes in a periodic field. Coupled with continuously designed GAN training procedures, IPFN is shown to synthesize tileable patterns with smooth transitions and local variations. Last but not least, thanks to both the adversarial training technique and the encoded Fourier features, IPFN learns high-frequency functions that produce authentic, high-quality results. To validate our approach, we present novel experimental results on various applications in 2D texture synthesis and 3D shape synthesis.

The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

The Bayesian paradigm has the potential to solve core issues of deep neural networks such as poor calibration and data inefficiency. Alas, scaling Bayesian inference to large weight spaces often requires restrictive approximations. In this work, we show that it suffices to perform inference over a small subset of model weights in order to obtain accurate predictive posteriors. The other weights are kept as point estimates. This subnetwork inference framework enables us to use expressive, otherwise intractable, posterior approximations over such subsets. In particular, we implement subnetwork linearized Laplace: We first obtain a MAP estimate of all weights and then infer a full-covariance Gaussian posterior over a subnetwork. We propose a subnetwork selection strategy that aims to maximally preserve the model's predictive uncertainty. Empirically, our approach is effective compared to ensembles and less expressive posterior approximations over full networks.