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Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.

We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Upcoming technologies like digital twins, autonomous, and artificial intelligent systems involving safety-critical applications require models which are accurate, interpretable, computationally efficient, and generalizable. Unfortunately, the two most commonly used modeling approaches, physics-based modeling (PBM) and data-driven modeling (DDM) fail to satisfy all these requirements. In the current work, we demonstrate how a hybrid approach combining the best of PBM and DDM can result in models which can outperform them both. We do so by combining partial differential equations based on first principles describing partially known physics with a black box DDM, in this case, a deep neural network model compensating for the unknown physics. First, we present a mathematical argument for why this approach should work and then apply the hybrid approach to model two dimensional heat diffusion problem with an unknown source term. The result demonstrates the method's superior performance in terms of accuracy, and generalizability. Additionally, it is shown how the DDM part can be interpreted within the hybrid framework to make the overall approach reliable.

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy. We demonstrate trapz-PiNNs have high expressive power through predicting solution with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as pointwise absolute and relative errors to analyze where could be further improved. We present an effective method for improving performance of trapz-PiNN on local metrics, provided that physical observations of high-fidelity simulation of the true solution are available. Besides the usual advantages of the deep learning solvers such as adaptivity and mesh-independence, the trapz-PiNN is able to solve PDEs with fractional laplacian with arbitrary $\alpha\in (0,2)$ and specializes on rectangular domain. It also has potential to be generalized into higher dimensions.

The architecture and the parameters of neural networks are often optimized independently, which requires costly retraining of the parameters whenever the architecture is modified. In this work we instead focus on growing the architecture without requiring costly retraining. We present a method that adds new neurons during training without impacting what is already learned, while improving the training dynamics. We achieve the latter by maximizing the gradients of the new weights and find the optimal initialization efficiently by means of the singular value decomposition (SVD). We call this technique Gradient Maximizing Growth (GradMax) and demonstrate its effectiveness in variety of vision tasks and architectures.

In this paper, we propose an analysis of the automorphism group of polar codes, with the scope of designing codes tailored for automorphism ensemble (AE) decoding. We prove the equivalence between the notion of decreasing monomial codes and the universal partial order (UPO) framework for the description of polar codes. Then, we analyze the algebraic properties of the affine automorphisms group of polar codes, providing a novel description of its structure and proposing a classification of automorphisms providing the same results under permutation decoding. Finally, we propose a method to list all the automorphisms that may lead to different candidates under AE decoding; by introducing the concept of redundant automorphisms, we find the maximum number of permutations providing possibly different codeword candidates under AE-SC, proposing a method to list all of them. A numerical analysis of the error correction performance of AE algorithm for the decoding of polar codes concludes the paper.

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

Deep Convolutional Neural Networks (CNNs) are a special type of Neural Networks, which have shown state-of-the-art results on various competitive benchmarks. The powerful learning ability of deep CNN is largely achieved with the use of multiple non-linear feature extraction stages that can automatically learn hierarchical representation from the data. Availability of a large amount of data and improvements in the hardware processing units have accelerated the research in CNNs and recently very interesting deep CNN architectures are reported. The recent race in deep CNN architectures for achieving high performance on the challenging benchmarks has shown that the innovative architectural ideas, as well as parameter optimization, can improve the CNN performance on various vision-related tasks. In this regard, different ideas in the CNN design have been explored such as use of different activation and loss functions, parameter optimization, regularization, and restructuring of processing units. However, the major improvement in representational capacity is achieved by the restructuring of the processing units. Especially, the idea of using a block as a structural unit instead of a layer is gaining substantial appreciation. This survey thus focuses on the intrinsic taxonomy present in the recently reported CNN architectures and consequently, classifies the recent innovations in CNN architectures into seven different categories. These seven categories are based on spatial exploitation, depth, multi-path, width, feature map exploitation, channel boosting and attention. Additionally, it covers the elementary understanding of the CNN components and sheds light on the current challenges and applications of CNNs.