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Most existing neural network-based approaches for solving stochastic optimal control problems using the associated backward dynamic programming principle rely on the ability to simulate the underlying state variables. However, in some problems, this simulation is infeasible, leading to the discretization of state variable space and the need to train one neural network for each data point. This approach becomes computationally inefficient when dealing with large state variable spaces. In this paper, we consider a class of this type of stochastic optimal control problems and introduce an effective solution employing multitask neural networks. To train our multitask neural network, we introduce a novel scheme that dynamically balances the learning across tasks. Through numerical experiments on real-world derivatives pricing problems, we prove that our method outperforms state-of-the-art approaches.

This paper introduces a novel multimodal and high-resolution human brain cerebellum lobule segmentation method. Unlike current tools that operate at standard resolution ($1 \text{ mm}^{3}$) or using mono-modal data, the proposed method improves cerebellum lobule segmentation through the use of a multimodal and ultra-high resolution ($0.125 \text{ mm}^{3}$) training dataset. To develop the method, first, a database of semi-automatically labelled cerebellum lobules was created to train the proposed method with ultra-high resolution T1 and T2 MR images. Then, an ensemble of deep networks has been designed and developed, allowing the proposed method to excel in the complex cerebellum lobule segmentation task, improving precision while being memory efficient. Notably, our approach deviates from the traditional U-Net model by exploring alternative architectures. We have also integrated deep learning with classical machine learning methods incorporating a priori knowledge from multi-atlas segmentation, which improved precision and robustness. Finally, a new online pipeline, named DeepCERES, has been developed to make available the proposed method to the scientific community requiring as input only a single T1 MR image at standard resolution.

Physics-informed neural networks (PINNs), rooted in deep learning, have emerged as a promising approach for solving partial differential equations (PDEs). By embedding the physical information described by PDEs into feedforward neural networks, PINNs are trained as surrogate models to approximate solutions without the need for label data. Nevertheless, even though PINNs have shown remarkable performance, they can face difficulties, especially when dealing with equations featuring rapidly changing solutions. These difficulties encompass slow convergence, susceptibility to becoming trapped in local minima, and reduced solution accuracy. To address these issues, we propose a binary structured physics-informed neural network (BsPINN) framework, which employs binary structured neural network (BsNN) as the neural network component. By leveraging a binary structure that reduces inter-neuron connections compared to fully connected neural networks, BsPINNs excel in capturing the local features of solutions more effectively and efficiently. These features are particularly crucial for learning the rapidly changing in the nature of solutions. In a series of numerical experiments solving Burgers equation, Euler equation, Helmholtz equation, and high-dimension Poisson equation, BsPINNs exhibit superior convergence speed and heightened accuracy compared to PINNs. From these experiments, we discover that BsPINNs resolve the issues caused by increased hidden layers in PINNs resulting in over-smoothing, and prevent the decline in accuracy due to non-smoothness of PDEs solutions.

This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.

A crucial challenge for solving problems in conflict research is in leveraging the semi-supervised nature of the data that arise. Observed response data such as counts of battle deaths over time indicate latent processes of interest such as intensity and duration of conflicts, but defining and labeling instances of these unobserved processes requires nuance and imprecision. The availability of such labels, however, would make it possible to study the effect of intervention-related predictors - such as ceasefires - directly on conflict dynamics (e.g., latent intensity) rather than through an intermediate proxy like observed counts of battle deaths. Motivated by this problem and the new availability of the ETH-PRIO Civil Conflict Ceasefires data set, we propose a Bayesian autoregressive (AR) hidden Markov model (HMM) framework as a sufficiently flexible machine learning approach for semi-supervised regime labeling with uncertainty quantification. We motivate our approach by illustrating the way it can be used to study the role that ceasefires play in shaping conflict dynamics. This ceasefires data set is the first systematic and globally comprehensive data on ceasefires, and our work is the first to analyze this new data and to explore the effect of ceasefires on conflict dynamics in a comprehensive and cross-country manner.

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.

The delimitation of biological species, i.e., deciding which individuals belong to the same species and whether and how many different species are represented in a data set, is key to the conservation of biodiversity. Much existing work uses only genetic data for species delimitation, often employing some kind of cluster analysis. This can be misleading, because geographically distant groups of individuals can be genetically quite different even if they belong to the same species. This paper investigates the problem of testing whether two potentially separated groups of individuals can belong to a single species or not based on genetic and spatial data. Various approaches are compared (some of which already exist in the literature) based on simulated metapopulations generated with SLiM and GSpace - two software packages that can simulate spatially-explicit genetic data at an individual level. Approaches involve partial Mantel testing, maximum likelihood mixed-effects models with a population effect, and jackknife-based homogeneity tests. A key challenge is that most tests perform on genetic and geographical distance data, violating standard independence assumptions. Simulations showed that partial Mantel tests and mixed-effects models have larger power than jackknife-based methods, but tend to display type-I-error rates slightly above the significance level. Moreover, a multiple regression model neglecting the dependence in the dissimilarities did not show inflated type-I-error rate. An application on brassy ringlets concludes the paper.

Recently, a family of unconventional integrators for ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for quadratic vector fields. All these integrators seem to possess remarkable conservation properties. In particular, it has been proved that, when the underlying ODE is Hamiltonian, its polarization discretization possesses an integral of motion and an invariant volume form. In this note, we propose a new algebraic approach to derivation of the integrals of motion for polarization discretizations.

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

Human cognition operates on a "Global-first" cognitive mechanism, prioritizing information processing based on coarse-grained details. This mechanism inherently possesses an adaptive multi-granularity description capacity, resulting in computational traits such as efficiency, robustness, and interpretability. The analysis pattern reliance on the finest granularity and single-granularity makes most existing computational methods less efficient, robust, and interpretable, which is an important reason for the current lack of interpretability in neural networks. Multi-granularity granular-ball computing employs granular-balls of varying sizes to daptively represent and envelop the sample space, facilitating learning based on these granular-balls. Given that the number of coarse-grained "granular-balls" is fewer than sample points, granular-ball computing proves more efficient. Moreover, the inherent coarse-grained nature of granular-balls reduces susceptibility to fine-grained sample disturbances, enhancing robustness. The multi-granularity construct of granular-balls generates topological structures and coarse-grained descriptions, naturally augmenting interpretability. Granular-ball computing has successfully ventured into diverse AI domains, fostering the development of innovative theoretical methods, including granular-ball classifiers, clustering techniques, neural networks, rough sets, and evolutionary computing. This has notably ameliorated the efficiency, noise robustness, and interpretability of traditional methods. Overall, granular-ball computing is a rare and innovative theoretical approach in AI that can adaptively and simultaneously enhance efficiency, robustness, and interpretability. This article delves into the main application landscapes for granular-ball computing, aiming to equip future researchers with references and insights to refine and expand this promising theory.