arXiv AI
Aug 3
A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem
arXiv:2607. 28733v1 Announce Type: cross Abstract: This proceedings contribution elaborates on the findings of arXiv:2605.
Minimal surfaces, Knots, and Neural Networks
arXiv:2605. 26234v2 Announce Type: replace-cross Abstract: A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number.
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arXiv Machine Learning
Jul 28
Neural Representation of Minimal Surfaces
arXiv:2607. 23437v1 Announce Type: cross Abstract: We propose a neural representation for minimal surfaces.
arXiv Machine Learning
Jun 18
A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces
arXiv:2606. 18759v1 Announce Type: cross Abstract: The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically.
arXiv Machine Learning
Aug 11
Eikonal Regularisation in Physics-Informed Neural Networks for Three-Dimensional Level-Set Advection: Transferability of Two-Dimensional Design Principles
arXiv:2608. 08322v1 Announce Type: cross Abstract: Physics-informed neural networks applied to the level-set formulation of interface advection commonly augment the residual and initial-condition losses with an eikonal regulariser, penalising the deviation of $\|\nabla\phi\|$ from unity.
arXiv AI
Jun 10
Moonshine: An Autonomous Mathematical Research Agent Centered on Conjecture Generation
arXiv:2606. 10806v1 Announce Type: new Abstract: Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures.
arXiv Machine Learning
Jul 1
Low-dimensional topology of deep neural networks
arXiv:2606. 31856v1 Announce Type: new Abstract: We study layered models, including feedforward networks, ResNets, and transformers, by limiting each layer to a width of $d = 3$, i.
arXiv Machine Learning
Jul 2
Geometry-Preserving Neural Architectures on Manifolds with Boundary
arXiv:2602. 03082v2 Announce Type: replace Abstract: A growing number of neural architectures have been proposed to enforce geometric constraints, including projection-based networks, exponential-map updates, constrained output layers, and manifold neural ODEs.
arXiv Machine Learning
Jul 3
An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
arXiv Machine Learning
Jul 31
Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries
arXiv:2605. 00760v2 Announce Type: replace Abstract: This paper deals with solving the 2D Helmholtz equation on non-parametric domains, leveraging a physics-informed neural operator network, the DeepONet framework.
arXiv Machine Learning
Jun 19
Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.
arXiv Machine Learning
Aug 4
Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions
arXiv:2508. 07559v3 Announce Type: replace-cross Abstract: We study the complexity of approximating high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube using Barron spaces.
arXiv Machine Learning
Jul 22
Hierarchical Physics-Embedded Learning for Partially Known Spatiotemporal Dynamics
arXiv:2510. 25306v3 Announce Type: replace Abstract: Partial physical knowledge--governing structures known, constitutive relations or their combinations not--pervades spatiotemporal systems.