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This paper makes two contributions to the field of text-based patent similarity. First, it compares the performance of different kinds of patent-specific pretrained embedding models, namely static word embeddings (such as word2vec and doc2vec models) and contextual word embeddings (such as transformers based models), on the task of patent similarity calculation. Second, it compares specifically the performance of Sentence Transformers (SBERT) architectures with different training phases on the patent similarity task. To assess the models' performance, we use information about patent interferences, a phenomenon in which two or more patent claims belonging to different patent applications are proven to be overlapping by patent examiners. Therefore, we use these interferences cases as a proxy for maximum similarity between two patents, treating them as ground-truth to evaluate the performance of the different embedding models. Our results point out that, first, Patent SBERT-adapt-ub, the domain adaptation of the pretrained Sentence Transformer architecture proposed in this research, outperforms the current state-of-the-art in patent similarity. Second, they show that, in some cases, large static models performances are still comparable to contextual ones when trained on extensive data; thus, we believe that the superiority in the performance of contextual embeddings may not be related to the actual architecture but rather to the way the training phase is performed.

We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

Deep generative models aim to learn the underlying distribution of data and generate new ones. Despite the diversity of generative models and their high-quality generation performance in practice, most of them lack rigorous theoretical convergence proofs. In this work, we aim to establish some convergence results for OT-Flow, one of the deep generative models. First, by reformulating the framework of OT-Flow model, we establish the $\Gamma$-convergence of the formulation of OT-flow to the corresponding optimal transport (OT) problem as the regularization term parameter $\alpha$ goes to infinity. Second, since the loss function will be approximated by Monte Carlo method in training, we established the convergence between the discrete loss function and the continuous one when the sample number $N$ goes to infinity as well. Meanwhile, the approximation capability of the neural network provides an upper bound for the discrete loss function of the minimizers. The proofs in both aspects provide convincing assurances for OT-Flow.

In the search for highly efficient decoders for short LDPC codes approaching maximum likelihood performance, a relayed decoding strategy, specifically activating the ordered statistics decoding process upon failure of a neural min-sum decoder, is enhanced by instilling three innovations. Firstly, soft information gathered at each step of the neural min-sum decoder is leveraged to forge a new reliability measure using a convolutional neural network. This measure aids in constructing the most reliable basis of ordered statistics decoding, bolstering the decoding process by excluding error-prone bits or concentrating them in a smaller area. Secondly, an adaptive ordered statistics decoding process is introduced, guided by a derived decoding path comprising prioritized blocks, each containing distinct test error patterns. The priority of these blocks is determined from the statistical data during the query phase. Furthermore, effective complexity management methods are devised by adjusting the decoding path's length or refining constraints on the involved blocks. Thirdly, a simple auxiliary criterion is introduced to reduce computational complexity by minimizing the number of candidate codewords before selecting the optimal estimate. Extensive experimental results and complexity analysis strongly support the proposed framework, demonstrating its advantages in terms of high throughput, low complexity, independence from noise variance, in addition to superior decoding performance.

Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.

Entropy conditions play a crucial role in the extraction of a physically relevant solution for a system of conservation laws, thus motivating the construction of entropy stable schemes that satisfy a discrete analogue of such conditions. TeCNO schemes (Fjordholm et al. 2012) form a class of arbitrary high-order entropy stable finite difference solvers, which require specialized reconstruction algorithms satisfying the sign property at each cell interface. Recently, third-order WENO schemes called SP-WENO (Fjordholm and Ray, 2016) and SP-WENOc (Ray, 2018) have been designed to satisfy the sign property. However, these WENO algorithms can perform poorly near shocks, with the numerical solutions exhibiting large spurious oscillations. In the present work, we propose a variant of the SP-WENO, termed as Deep Sign-Preserving WENO (DSP-WENO), where a neural network is trained to learn the WENO weighting strategy. The sign property and third-order accuracy are strongly imposed in the algorithm, which constrains the WENO weight selection region to a convex polygon. Thereafter, a neural network is trained to select the WENO weights from this convex region with the goal of improving the shock-capturing capabilities without sacrificing the rate of convergence in smooth regions. The proposed synergistic approach retains the mathematical framework of the TeCNO scheme while integrating deep learning to remedy the computational issues of the WENO-based reconstruction. We present several numerical experiments to demonstrate the significant improvement with DSP-WENO over the existing variants of WENO satisfying the sign property.

We identify reduced order models (ROM) of forced systems from data using invariant foliations. The forcing can be external, parametric, periodic or quasi-periodic. The process has four steps: 1. identify an approximate invariant torus and the linear dynamics about the torus; 2. identify a globally defined invariant foliation about the torus; 3. identify a local foliation about an invariant manifold that complements the global foliation 4. extract the invariant manifold as the leaf going through the torus and interpret the result. We combine steps 2 and 3, so that we can track the location of the invariant torus and scale the invariance equations appropriately. We highlight some fundamental limitations of invariant manifolds and foliations when fitting them to data, that require further mathematics to resolve.

In many application settings, the data have missing entries which make analysis challenging. An abundant literature addresses missing values in an inferential framework: estimating parameters and their variance from incomplete tables. Here, we consider supervised-learning settings: predicting a target when missing values appear in both training and testing data. We show the consistency of two approaches in prediction. A striking result is that the widely-used method of imputing with a constant, such as the mean prior to learning is consistent when missing values are not informative. This contrasts with inferential settings where mean imputation is pointed at for distorting the distribution of the data. That such a simple approach can be consistent is important in practice. We also show that a predictor suited for complete observations can predict optimally on incomplete data, through multiple imputation. Finally, to compare imputation with learning directly with a model that accounts for missing values, we analyze further decision trees. These can naturally tackle empirical risk minimization with missing values, due to their ability to handle the half-discrete nature of incomplete variables. After comparing theoretically and empirically different missing values strategies in trees, we recommend using the "missing incorporated in attribute" method as it can handle both non-informative and informative missing values.

We present a novel framework for the development of fourth-order lattice Boltzmann schemes to tackle multidimensional nonlinear systems of conservation laws. Our numerical schemes preserve two fundamental characteristics inherent in classical lattice Boltzmann methods: a local relaxation phase and a transport phase composed of elementary shifts on a Cartesian grid. Achieving fourth-order accuracy is accomplished through the composition of second-order time-symmetric basic schemes utilizing rational weights. This enables the representation of the transport phase in terms of elementary shifts. Introducing local variations in the relaxation parameter during each stage of relaxation ensures the entropic nature of the schemes. This not only enhances stability in the long-time limit but also maintains fourth-order accuracy. To validate our approach, we conduct comprehensive testing on scalar equations and systems in both one and two spatial dimensions.

We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.