arXiv:2607. 23437v1 Announce Type: cross Abstract: We propose a neural representation for minimal 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:2607. 28733v1 Announce Type: cross Abstract: This proceedings contribution elaborates on the findings of arXiv:2605.
arXiv:2608.15933v1 Announce Type: cross Abstract: Implicit surface representations have regained popularity because of their use in machine learning. A common component in optimization is regularizat...
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.
arXiv:2607. 15751v1 Announce Type: new Abstract: This work introduces Physics-Informed Splines (PI-Splines), a structured spline-based architecture for physics-informed learning.
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.
The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.
The paper introduces a linearized Physics-Informed Neural Network (lPINN), a reduced‑order neural basis approach for solving forward and inverse differential equations. In an offline phase, lPINN learns continuous, differentiable neural basis functions from numerical solutions, which are then frozen for new problem instances; the online solution is obtained by minimizing the governing‑equation residual with additional constraints. Experiments on advection‑diffusion, Burgers', and nonlinear pendulum equations show that lPINN achieves lower solution and parameter errors than vanilla PINNs while reducing online inference times by up to three orders of magnitude, and its continuous representation generalizes to finer meshes without retraining.
arXiv:2606. 04736v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) have become a promising framework for simulating partial differential equations (PDEs) by embedding physical laws directly into neural network training.
The paper introduces a hybrid iterative deep Ritz method (H-IDRM) for solving interface problems involving second-order elliptic operators. It uses a mixed formulation that reduces the problem to a sequence of convex minimization tasks and employs a level‑set neural network to represent the interface, thereby handling piecewise smooth solutions without explicit interface sampling. The authors analyze errors from neural network, Monte Carlo, iterative, and penalty sources, and demonstrate through numerical experiments that H-IDRM outperforms existing neural solvers on high‑dimensional, complex interface problems.
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:2609.36806v1 Announce Type: new Abstract: Modern engineering systems, from automobiles to aircraft, are designed by using precise, continuous parametric computer-aided design (CAD) models. Eval...