arXiv:2604. 01275v3 Announce Type: replace-cross Abstract: In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization.
arXiv:2609. 23529v1 Announce Type: new Abstract: Neural operators have emerged as powerful surrogates for solving partial differential equations (PDEs), yet their reliability under distribution shift remains a critical barrier to deployment.
The paper introduces Spectral-like Neural Discretisation (SpeND), a mesh‑free method that learns stencil weights via a neural network to approximate the modal response of a spectral operator across a specified band of wavenumbers. By projecting the network output onto the space of polynomial‑consistent weights, SpeND ensures exact consistency while minimizing dispersion and dissipation errors in a self‑supervised, physics‑agnostic manner. Experiments on disordered 2‑D node sets demonstrate that the learned fourth‑order operator matches the exact spectral response over a wider band than traditional LABFM or structured‑grid finite differences, and retains fourth‑order convergence upon refinement.
arXiv:2606. 31915v1 Announce Type: cross Abstract: While conformal prediction provides a general framework for uncertainty quantification in predictive inference, its application is often limited by computational cost.
The paper introduces Spectrally Optimised Neural Discretisations (SpeND), a mesh‑free framework that learns discretisation weights from local stencil geometry on unstructured point clouds. By embedding discrete moment conditions into the network architecture, SpeND guarantees polynomial consistency and allows the weights to be optimised for spectral accuracy over a chosen wavenumber band, using an unsupervised Fourier‑mode loss. The resulting operators are PDE‑agnostic, perform well on Poisson, Burgers, and Navier–Stokes equations, and can reduce wall‑clock time by 3–20× compared to existing mesh‑free methods at the same accuracy.
arXiv:2602. 09530v2 Announce Type: replace-cross Abstract: We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization.
The paper explores multivariate conformal uncertainty propagation for multitask atomistic simulations, introducing methods such as Bonferroni‑corrected hyperrectangles, hyperellipsoidal sets based on Mahalanobis distance, and custom loss functions within conformal risk control. It applies these techniques to calibrate predictions of energies, forces, and stresses, then propagates the resulting uncertainty sets to downstream quantities like elastic constants and vacancy formation energies. The study emphasizes how incorporating correlation predictions can capture symmetries and error cancellation, and discusses the interaction between computational protocols and conformal guarantees.
arXiv:2605. 20440v2 Announce Type: replace Abstract: Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude.
arXiv:2609.36047v1 Announce Type: cross Abstract: Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional....
arXiv:2606. 00401v1 Announce Type: cross Abstract: Simulating large molecular systems comprising thousands of atoms requires highly scalable methodologies.
arXiv:2606. 08654v1 Announce Type: new Abstract: In this paper, we propose a perturbation-based conformal prediction framework for uncertainty quantification in operator learning, with a focus on the 2D Navier--Stokes equations.
arXiv:2402. 16388v4 Announce Type: replace-cross Abstract: The need for uncertainty quantification in anomaly detection systems has become increasingly important.
arXiv:2606. 13827v1 Announce Type: cross Abstract: Markovian Whittle-Mat\'ern fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (\kappa^2 - \Delta)^{\alpha/2} u = \mathcal{W}, \;\; \kappa \in \mathbb{R}, \; \alpha \in \mathbb{N}.