The paper tackles the challenge of predicting multiple high‑dimensional physical fields that must satisfy linear equality constraints, a common scenario in physics‑informed machine learning. It critiques the conventional approach of deducing one field from others, showing its sensitivity to arbitrary choices and its impact on accuracy and uncertainty. To address this, the authors introduce a symmetric framework that first applies a row‑wise PCA to preserve constraints in a latent space, then trains a linearly‑constrained multi‑output Gaussian process using a specially parametrized kernel, and validate the method on population dynamics and CFD problems involving Reynolds stress tensors.
By Mahamat Hamdan Nassouradine, Cl\'ement Gauchy, Pierre-Emmanuel Angeli, S\'ebastien da Veiga
arXiv:2608.24386v1 Announce Type: cross
Abstract: Tensor-valued prediction is fundamental to geometric deep learning, yet uncertainty quantification (UQ) for such outputs remains an open challenge. W...
By Ruihan Liu, Yu Ji, Jianbo Yu, Shifu Yan, Qingchao Jiang
arXiv:2603. 22050v2 Announce Type: replace-cross Abstract: Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data.
By Atticus Rex, Elizabeth Qian, David Peterson
arXiv:2606. 14053v1 Announce Type: cross Abstract: Quantifying the influence of hybrid aleatory and epistemic uncertainties on high-dimensional system responses remains a major challenge in global sensitivity analysis (GSA).
By Shijie Zhong, Jiangfeng Fu, Pengfei Wei
arXiv:2607. 05025v1 Announce Type: new Abstract: Vibration-based damage identification in civil infrastructure is a challenging, ill-posed inverse problem due to measurement noise, sparse sensor arrays, and environmental variability.
By Ana Fernandez Navamuel, A. Javier Omella, Diego Zamora-Sanchez, David Pardo
arXiv:2512.12749v3 Announce Type: replace-cross
Abstract: Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural opera...
By Sahil Bhola, Karthik Duraisamy
arXiv:2608. 11435v1 Announce Type: new Abstract: Forward and inverse modeling of parametric dynamical systems requires surrogate models that are not only accurate for state prediction, but also informative for parameter calibration.
By Qiyao Zhou, Xujia Zhu, Pierre Joli, Yu Cong, Sibo Cheng
arXiv:2609. 21085v1 Announce Type: cross Abstract: Gaussian processes (GPs) provide principled probabilistic predictions while encoding prior knowledge, including equivariances.
By Tim Steinert, David Ginsbourger
arXiv:2608.24518v1 Announce Type: new
Abstract: Uncertainty Quantification (UQ) plays a vital role in enhancing the reliability of deep learning model predictions, especially in scenarios with high-d...
By Leonhard F. Feiner, Manuel Nickel, Martin Menten, Laurin Lux, Rickmer Braren, Daniel Rueckert, Georgios Kaissis, Raphael Rehms, Johannes Paetzold
arXiv:2603. 29237v2 Announce Type: replace Abstract: Enforcing prescribed global integral constraints in mesh-free neural PDE solvers is challenging in high-dimensional domains.
By Zhangyong Liang, Huanhuan Gao
Uncertainty Quantification (UQ) plays a vital role in enhancing the reliability of deep learning model predictions, especially in scenarios with high-dimensional output spaces. This paper addresses th...
The paper introduces a new representation‑adaptive kernel class that, on a fixed sample, yields a union of reproducing‑kernel Hilbert‑space ellipsoids instead of a single ellipsoid. It defines a minimum‑trace common covariance dominating the empirical union generated by Brownian kernel ladders, and derives exact formulations, statistical and computational consequences, and a universal Gaussian‑complexity bound. The work further develops geometric reductions, deterministic depth laws, and exact empirical Kolmogorov‑width formulas, providing both lower and upper certificates for covariance certification and illustrating the distinction between successful covariance certification and predictive selection.
By Mahdi Mohammadigohari