A Neural Network Framework for Geodesic-Like Curve Computation on Parametric Surfaces

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss landscapes. We propose a lightweight curvature-aware optimization framework that augments existing first-order optimizers with an adaptive predictive correction based on secant information. Consecutive gradient differences are used as a cheap proxy for local geometric change, together with a step-normalized secant curvature indicator to control the correction strength. The framework is plug-and-play, computationally efficient, and broadly compatible with existing optimizers, without explicitly forming second-order matrices. Experiments on diverse PDE benchmarks show consistent improvements in convergence speed, training stability, and solution accuracy over standard optimizers and strong baselines, including on the high-dimensional heat equation, Gray--Scott system, Belousov--Zhabotinsky system, and 2D Kuramoto--Sivashinsky system.

We propose a neural representation for minimal surfaces. Unlike prior approaches based on discretization or Physics-Informed Neural Networks (PINNs), where meshes or neural fields are optimized to approximate the governing equations, our method builds on an exact representation, similar to the classical Weierstrass--Enneper parameterization, yielding minimal surfaces up to negligible quadrature error in evaluation. We formulate a training objective for the Plateau problem that optimizes over this representation.

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. However, recent studies show that PINN optimisation is sensitive to numerical precision. Existing implementations commonly use either single precision (FP32), which is computationally efficient but prone to failure modes, or double precision (FP64), which is robust but substantially expensive. This creates a trade-off between computational efficiency and numerical accuracy. To reduce the computational cost of double-precision training while retaining prediction accuracy, we propose a curvature-aware precision controller that adapts numerical precision during training rather than treating it as a fixed implementation choice. The proposed method reuses curvature information derived from the limited-memory BFGS (L-BFGS) optimiser to construct a precision controller, retaining FP32 when lower precision is sufficient and promoting computation to FP64 when the training dynamics indicate numerical sensitivity or precision-limited stagnation. We evaluate the proposed approach on four canonical PINN failure-mode benchmarks and an irradiance-driven ordinary differential equation example. We further test the proposed approach across different neural network architectures. The method consistently matches or even slightly exceeds full FP64 solution accuracy while reducing training time relative to full double-precision training on all benchmark equations. The obtained results indicate that precision sensitivity in PINN optimisation is phase-dependent, and that selectively applying higher precision only during numerically critical stages can lower computational cost without sacrificing predictive accuracy.