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We present an algorithm for computing melting points by autonomously learning from coexistence simulations in the NPT ensemble. Given the interatomic interaction model, the method makes decisions regarding the number of atoms and temperature at which to conduct simulations, and based on the collected data predicts the melting point along with the uncertainty, which can be systematically improved with more data. We demonstrate how incorporating physical models of the solid-liquid coexistence evolution enhances the algorithm's accuracy and enables optimal decision-making to effectively reduce predictive uncertainty. To validate our approach, we compare the results of 20 melting point calculations from the literature to the results of our calculations, all conducted with same interatomic potentials. Remarkably, we observe significant deviations in about one-third of the cases, underscoring the need for accurate and reliable algorithms for materials property calculations.

Linear logic has provided new perspectives on proof-theory, denotational semantics and the study of programming languages. One of its main successes are proof-nets, canonical representations of proofs that lie at the intersection between logic and graph theory. In the case of the minimalist proof-system of multiplicative linear logic without units (MLL), these two aspects are completely fused: proof-nets for this system are graphs satisfying a correctness criterion that can be fully expressed in the language of graphs. For more expressive logical systems (containing logical constants, quantifiers and exponential modalities), this is not completely the case. The purely graphical approach of proof-nets deprives them of any sequential structure that is crucial to represent the order in which arguments are presented, which is necessary for these extensions. Rebuilding this order of presentation - sequentializing the graph - is thus a requirement for a graph to be logical. Presentations and study of the artifacts ensuring that sequentialization can be done, such as boxes or jumps, are an integral part of researches on linear logic. Jumps, extensively studied by Faggian and di Giamberardino, can express intermediate degrees of sequentialization between a sequent calculus proof and a fully desequentialized proof-net. We propose to analyze the logical strength of jumps by internalizing them in an extention of MLL where axioms on a specific formula, the jumping formula, introduce constrains on the possible sequentializations. The jumping formula needs to be treated non-linearly, which we do either axiomatically, or by embedding it in a very controlled fragment of multiplicative-exponential linear logic, uncovering the exponential logic of sequentialization.

In this paper, by using methods of $D$-companion matrix, we reprove a generalization of the Guass-Lucas theorem and get the majorization relationship between the zeros of convex combinations of incomplete polynomials and an origin polynomial. Moreover, we prove that the set of all zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. The location of zeros of convex combinations of incomplete polynomials is determined.

Complexity is a fundamental concept underlying statistical learning theory that aims to inform generalization performance. Parameter count, while successful in low-dimensional settings, is not well-justified for overparameterized settings when the number of parameters is more than the number of training samples. We revisit complexity measures based on Rissanen's principle of minimum description length (MDL) and define a novel MDL-based complexity (MDL-COMP) that remains valid for overparameterized models. MDL-COMP is defined via an optimality criterion over the encodings induced by a good Ridge estimator class. We provide an extensive theoretical characterization of MDL-COMP for linear models and kernel methods and show that it is not just a function of parameter count, but rather a function of the singular values of the design or the kernel matrix and the signal-to-noise ratio. For a linear model with $n$ observations, $d$ parameters, and i.i.d. Gaussian predictors, MDL-COMP scales linearly with $d$ when $d<n$, but the scaling is exponentially smaller -- $\log d$ for $d>n$. For kernel methods, we show that MDL-COMP informs minimax in-sample error, and can decrease as the dimensionality of the input increases. We also prove that MDL-COMP upper bounds the in-sample mean squared error (MSE). Via an array of simulations and real-data experiments, we show that a data-driven Prac-MDL-COMP informs hyper-parameter tuning for optimizing test MSE with ridge regression in limited data settings, sometimes improving upon cross-validation and (always) saving computational costs. Finally, our findings also suggest that the recently observed double decent phenomenons in overparameterized models might be a consequence of the choice of non-ideal estimators.

Physics informed neural network (PINN) based solution methods for differential equations have recently shown success in a variety of scientific computing applications. Several authors have reported difficulties, however, when using PINNs to solve equations with multiscale features. The objective of the present work is to illustrate and explain the difficulty of using standard PINNs for the particular case of divergence-form elliptic partial differential equations (PDEs) with oscillatory coefficients present in the differential operator. We show that if the coefficient in the elliptic operator $a^{\epsilon}(x)$ is of the form $a(x/\epsilon)$ for a 1-periodic coercive function $a(\cdot)$, then the Frobenius norm of the neural tangent kernel (NTK) matrix associated to the loss function grows as $1/\epsilon^2$. This implies that as the separation of scales in the problem increases, training the neural network with gradient descent based methods to achieve an accurate approximation of the solution to the PDE becomes increasingly difficult. Numerical examples illustrate the stiffness of the optimization problem.

We present a polymorphic linear lambda-calculus as a proof language for second-order intuitionistic linear logic. The calculus includes addition and scalar multiplication, enabling the proof of a linearity result at the syntactic level.

This study proposes an interpretable neural network-based non-proportional odds model (N$^3$POM) for ordinal regression. N$^3$POM is different from conventional approaches to ordinal regression with non-proportional models in several ways: (1) N$^3$POM is designed to directly handle continuous responses, whereas standard methods typically treat de facto ordered continuous variables as discrete, (2) instead of estimating response-dependent finite coefficients of linear models from discrete responses as is done in conventional approaches, we train a non-linear neural network to serve as a coefficient function. Thanks to the neural network, N$^3$POM offers flexibility while preserving the interpretability of conventional ordinal regression. We establish a sufficient condition under which the predicted conditional cumulative probability locally satisfies the monotonicity constraint over a user-specified region in the covariate space. Additionally, we provide a monotonicity-preserving stochastic (MPS) algorithm for effectively training the neural network. We apply N$^3$POM to several real-world datasets.

We study the problem of adaptive variable selection in a Gaussian white noise model of intensity $\varepsilon$ under certain sparsity and regularity conditions on an unknown regression function $f$. The $d$-variate regression function $f$ is assumed to be a sum of functions each depending on a smaller number $k$ of variables ($1 \leq k \leq d$). These functions are unknown to us and only few of them are non-zero. We assume that $d=d_\varepsilon \to \infty$ as $\varepsilon \to 0$ and consider the cases when $k$ is fixed and when $k=k_\varepsilon \to \infty$ and $k=o(d)$ as $\varepsilon \to 0$. In this work, we introduce an adaptive selection procedure that, under some model assumptions, identifies exactly all non-zero $k$-variate components of $f$. In addition, we establish conditions under which exact identification of the non-zero components is impossible. These conditions ensure that the proposed selection procedure is the best possible in the asymptotically minimax sense with respect to the Hamming risk.

Prognostic Health Management aims to predict the Remaining Useful Life (RUL) of degrading components/systems utilizing monitoring data. These RUL predictions form the basis for optimizing maintenance planning in a Predictive Maintenance (PdM) paradigm. We here propose a metric for assessing data-driven prognostic algorithms based on their impact on downstream PdM decisions. The metric is defined in association with a decision setting and a corresponding PdM policy. We consider two typical PdM decision settings, namely component ordering and/or replacement planning, for which we investigate and improve PdM policies that are commonly utilized in the literature. All policies are evaluated via the data-based estimation of the long-run expected maintenance cost per unit time, using monitored run-to-failure experiments. The policy evaluation enables the estimation of the proposed metric. We employ the metric as an objective function for optimizing heuristic PdM policies and algorithms' hyperparameters. The effect of different PdM policies on the metric is initially investigated through a theoretical numerical example. Subsequently, we employ four data-driven prognostic algorithms on a simulated turbofan engine degradation problem, and investigate the joint effect of prognostic algorithm and PdM policy on the metric, resulting in a decision-oriented performance assessment of these algorithms.

Signal detection is one of the main challenges of data science. As it often happens in data analysis, the signal in the data may be corrupted by noise. There is a wide range of techniques aimed at extracting the relevant degrees of freedom from data. However, some problems remain difficult. It is notably the case of signal detection in almost continuous spectra when the signal-to-noise ratio is small enough. This paper follows a recent bibliographic line which tackles this issue with field-theoretical methods. Previous analysis focused on equilibrium Boltzmann distributions for some effective field representing the degrees of freedom of data. It was possible to establish a relation between signal detection and $\mathbb{Z}_2$-symmetry breaking. In this paper, we consider a stochastic field framework inspiring by the so-called "Model A", and show that the ability to reach or not an equilibrium state is correlated with the shape of the dataset. In particular, studying the renormalization group of the model, we show that the weak ergodicity prescription is always broken for signals small enough, when the data distribution is close to the Marchenko-Pastur (MP) law. This, in particular, enables the definition of a detection threshold in the regime where the signal-to-noise ratio is small enough.