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
Aug 20
Learning Topological Features of $\widehat Z$-invariants
arXiv:2608. 18570v1 Announce Type: cross Abstract: Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology.
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
Sep 15
Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradients
arXiv:2609.14841v1 Announce Type: new Abstract: Scientific machine learning methods such as physics-informed neural networks (PINNs) increasingly rely on domain decomposition for better scalability w...
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
Sep 22
Learning Physics from an Imperfect Ancestor
arXiv:2609.24947v1 Announce Type: new Abstract: Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed n...
arXiv Machine Learning
2d ago
The limits of exactness: On the failure of automatic differentiation in physics-informed machine learning
arXiv:2609.33078v2 Announce Type: replace Abstract: Automatic differentiation (AD) lets neural networks compute derivatives of governing equations to machine precision, and this precision has made it...