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In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Data-driven research in Additive Manufacturing (AM) has gained significant success in recent years. This has led to a plethora of scientific literature to emerge. The knowledge in these works consists of AM and Artificial Intelligence (AI) contexts that have not been mined and formalized in an integrated way. Moreover, no tools or guidelines exist to support data-driven knowledge transfer from one context to another. As a result, data-driven solutions using specific AI techniques are being developed and validated only for specific AM process technologies. There is a potential to exploit the inherent similarities across various AM technologies and adapt the existing solutions from one process or problem to another using AI, such as Transfer Learning. We propose a three-step knowledge transferability analysis framework in AM to support data-driven AM knowledge transfer. As a prerequisite to transferability analysis, AM knowledge is featurized into identified knowledge components. The framework consists of pre-transfer, transfer, and post-transfer steps to accomplish knowledge transfer. A case study is conducted between flagship metal AM processes. Laser Powder Bed Fusion (LPBF) is the source of knowledge motivated by its relative matureness in applying AI over Directed Energy Deposition (DED), which drives the need for knowledge transfer as the less explored target process. We show successful transfer at different levels of the data-driven solution, including data representation, model architecture, and model parameters. The pipeline of AM knowledge transfer can be automated in the future to allow efficient cross-context or cross-process knowledge exchange.

This article presents a new tool for the automatic detection of meteors. Fast Meteor Detection Toolbox (FMDT) is able to detect meteor sightings by analyzing videos acquired by cameras onboard weather balloons or within airplane with stabilization. The challenge consists in designing a processing chain composed of simple algorithms, that are robust to the high fluctuation of the videos and that satisfy the constraints on power consumption (10 W) and real-time processing (25 frames per second).

It is crucial to detect when an instance lies downright too far from the training samples for the machine learning model to be trusted, a challenge known as out-of-distribution (OOD) detection. For neural networks, one approach to this task consists of learning a diversity of predictors that all can explain the training data. This information can be used to estimate the epistemic uncertainty at a given newly observed instance in terms of a measure of the disagreement of the predictions. Evaluation and certification of the ability of a method to detect OOD require specifying instances which are likely to occur in deployment yet on which no prediction is available. Focusing on regression tasks, we choose a simple yet insightful model for this OOD distribution and conduct an empirical evaluation of the ability of various methods to discriminate OOD samples from the data. Moreover, we exhibit evidence that a diversity of parameters may fail to translate to a diversity of predictors. Based on the choice of an OOD distribution, we propose a new way of estimating the entropy of a distribution on predictors based on nearest neighbors in function space. This leads to a variational objective which, combined with the family of distributions given by a generative neural network, systematically produces a diversity of predictors that provides a robust way to detect OOD samples.

There are now many options for doubly robust estimation; however, there is a concerning trend in the applied literature to believe that the combination of a propensity score and an adjusted outcome model automatically results in a doubly robust estimator and/or to misuse more complex established doubly robust estimators. A simple alternative, canonical link generalized linear models (GLM) fit via inverse probability of treatment (propensity score) weighted maximum likelihood estimation followed by standardization (the g-formula) for the average causal effect, is a doubly robust estimation method. Our aim is for the reader not just to be able to use this method, which we refer to as IPTW GLM, for doubly robust estimation, but to fully understand why it has the doubly robust property. For this reason, we define clearly, and in multiple ways, all concepts needed to understand the method and why it is doubly robust. In addition, we want to make very clear that the mere combination of propensity score weighting and an adjusted outcome model does not generally result in a doubly robust estimator. Finally, we hope to dispel the misconception that one can adjust for residual confounding remaining after propensity score weighting by adjusting in the outcome model for what remains `unbalanced' even when using doubly robust estimators. We provide R code for our simulations and real open-source data examples that can be followed step-by-step to use and hopefully understand the IPTW GLM method. We also compare to a much better-known but still simple doubly robust estimator.

Permutation tests are widely recognized as robust alternatives to tests based on the normal theory. Random permutation tests have been frequently employed to assess the significance of variables in linear models. Despite their widespread use, existing random permutation tests lack finite-sample and assumption-free guarantees for controlling type I error in partial correlation tests. To address this standing challenge, we develop a conformal test through permutation-augmented regressions, which we refer to as PALMRT. PALMRT not only achieves power competitive with conventional methods but also provides reliable control of type I errors at no more than $2\alpha$ given any targeted level $\alpha$, for arbitrary fixed-designs and error distributions. We confirmed this through extensive simulations. Compared to the cyclic permutation test (CPT), which also offers theoretical guarantees, PALMRT does not significantly compromise power or set stringent requirements on the sample size, making it suitable for diverse biomedical applications. We further illustrate their differences in a long-Covid study where PALMRT validated key findings previously identified using the t-test, while CPT suffered from a drastic loss of power. We endorse PALMRT as a robust and practical hypothesis test in scientific research for its superior error control, power preservation, and simplicity.

The problem of recovering partial derivatives of high orders of bivariate functions with finite smoothness is studied. Based on the truncation method, a numerical differentiation algorithm was constructed, which is optimal by the order, both in the sense of accuracy and in the sense of the amount of Galerkin information involved. Numerical demonstrations are provided to illustrate that the proposed method can be implemented successfully.

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

The prevalence of mobility impairments due to conditions such as spinal cord injuries, strokes, and degenerative diseases is on the rise globally. Lower-limb exoskeletons have been increasingly recognized as a viable solution for enhancing mobility and rehabilitation for individuals with such impairments. However, existing exoskeleton control systems often suffer from limitations such as latency, lack of adaptability, and computational inefficiency. To address these challenges, this paper introduces a novel online adversarial learning architecture integrated with edge computing for high-level lower-limb exoskeleton control. In the proposed architecture, sensor data from the user is processed in real-time through edge computing nodes, which then interact with an online adversarial learning model. This model adapts to the user's specific needs and controls the exoskeleton with minimal latency. Experimental evaluations demonstrate significant improvements in control accuracy and adaptability, as well as enhanced quality-of-service (QoS) metrics. These findings indicate that the integration of online adversarial learning with edge computing offers a robust and efficient approach for the next generation of lower-limb exoskeleton control systems.

Marginal structural models (MSMs) are often used to estimate causal effects of treatments on survival time outcomes from observational data when time-dependent confounding may be present. They can be fitted using, e.g., inverse probability of treatment weighting (IPTW). It is important to evaluate the performance of statistical methods in different scenarios, and simulation studies are a key tool for such evaluations. In such simulation studies, it is common to generate data in such a way that the model of interest is correctly specified, but this is not always straightforward when the model of interest is for potential outcomes, as is an MSM. Methods have been proposed for simulating from MSMs for a survival outcome, but these methods impose restrictions on the data-generating mechanism. Here we propose a method that overcomes these restrictions. The MSM can be a marginal structural logistic model for a discrete survival time or a Cox or additive hazards MSM for a continuous survival time. The hazard of the potential survival time can be conditional on baseline covariates, and the treatment variable can be discrete or continuous. We illustrate the use of the proposed simulation algorithm by carrying out a brief simulation study. This study compares the coverage of confidence intervals calculated in two different ways for causal effect estimates obtained by fitting an MSM via IPTW.