March 2026

arXiv:2603.24704v1 Announce Type: cross Abstract: In deploying artificial intelligence (AI) models, selective prediction offers the option to abstain from making a prediction when uncertain about model quality. To fulfill its promise, it is crucial to enforce strict and precise error control over cases where the model is trusted. We propose Selective Conformal Risk control with E-values (SCoRE), a new framework for deriving such decisions for any trained model and any user-defined, bounded and continuously-valued risk. SCoRE offers two […]

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arXiv:2603.24638v1 Announce Type: cross Abstract: The requirement of generating predictions that exactly fulfill the fundamental symmetry of the corresponding physical quantities has profoundly shaped the development of machine-learning models for physical simulations. In many cases, models are built using constrained mathematical forms that ensure that symmetries are enforced exactly. However, unconstrained models that do not obey rotational symmetries are often found to have competitive performance, and to be able to emph{learn} to a high level of accuracy an […]

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arXiv:2603.24613v1 Announce Type: cross Abstract: Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using […]

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arXiv:2603.25579v1 Announce Type: new Abstract: A key capability of modern neural networks is their capacity to simultaneously learn underlying rules and memorize specific facts or exceptions. Yet, theoretical understanding of this dual capability remains limited. We introduce the Rules-and-Facts (RAF) model, a minimal solvable setting that enables precise characterization of this phenomenon by bridging two classical lines of work in the statistical physics of learning: the teacher-student framework for generalization and Gardner-style capacity analysis for memorization. In the […]

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arXiv:2603.25530v1 Announce Type: new Abstract: Tensors provide a structured representation for multidimensional data, yet discretization can obscure important information when such data originates from continuous processes. We address this limitation by introducing a functional Tucker decomposition (FTD) that embeds mode-wise continuity constraints directly into the decomposition. The FTD employs reproducing kernel Hilbert spaces (RKHS) to model continuous modes without requiring an a-priori basis, while preserving the multi-linear subspace structure of the Tucker model. Through RKHS-driven representation, the model […]

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arXiv:2603.25466v1 Announce Type: new Abstract: We study statistical estimation in a student–teacher setting, where predictions from a pre-trained teacher are used to guide a student model. A standard approach is to train the student to directly match the teacher’s outputs, which we refer to as student soft matching (SM). This approach directly propagates any systematic bias or mis-specification present in the teacher, thereby degrading the student’s predictions. We propose and analyze an alternative scheme, known as residual-as-teacher (RaT), […]

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arXiv:2603.25370v1 Announce Type: new Abstract: Probabilistic forecasting provides a principled framework for uncertainty quantification in dynamical systems by representing predictions as probability distributions rather than deterministic trajectories. However, existing forecasting approaches, whether physics-based or neural-network-based, remain fundamentally trajectory-oriented: predictive distributions are usually accessed through ensembles or sampling, rather than evolved directly as dynamical objects. A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions. The framework introduces a distributional encoding and decoding structure […]

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arXiv:2603.25311v1 Announce Type: new Abstract: Efficient global optimization (EGO) is one of the most widely used noise-free Bayesian optimization algorithms.It comprises the Gaussian process (GP) surrogate model and expected improvement (EI) acquisition function. In practice, when EGO is applied, a scalar matrix of a small positive value (also called a nugget or jitter) is usually added to the covariance matrix of the deterministic GP to improve numerical stability. We refer to this EGO with a positive nugget as […]

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arXiv:2603.25224v1 Announce Type: new Abstract: Demographic parity (DP) is a widely used group fairness criterion requiring predictive distributions to be invariant across sensitive groups. While natural in classification, full distributional DP is often overly restrictive in regression and can lead to substantial accuracy loss. We propose a relaxation of DP tailored to regression, enforcing parity only at a finite set of quantile levels and/or score thresholds. Concretely, we introduce a novel (${ell}$, Z)-fair predictor, which imposes groupwise CDF […]

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arXiv:2603.25024v1 Announce Type: new Abstract: As a representative continuous-depth neural network approach, stochastic differential equation (SDE)-based Bayesian neural networks (BNNs) have attracted considerable attention due to their solid theoretical foundations and strong potential for real-world applications. However, their reliance on numerical SDE solvers inevitably incurs a large number of function evaluations (NFEs), resulting in high computational cost and occasional convergence instability. To address these challenges, we propose a Nesterov-accelerated gradient (NAG) enhanced SDE-BNN model. By integrating NAG into […]

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