PINNs in More General Geometry

2026-04-27

AI Summary

This work addresses the variational minimization of complex functionals in differential geometry by systematically introducing physics-informed neural networks (PINNs) into a general differential geometric framework for the first time. By directly embedding geometric functionals into the loss function of neural networks, the approach enables end-to-end numerical solutions without requiring mesh discretization. A rigorous correspondence is established between geometric variational problems and deep learning optimization objectives, allowing the method to effectively handle high-dimensional and nonlinear geometric structures. The study reproduces and integrates three representative works, demonstrating the feasibility and advantages of PINNs in solving intricate geometric variational problems, thereby offering a novel paradigm at the intersection of differential geometry and artificial intelligence.

Abstract

Neural architectures trained with losses inspired by differential conditions are the basis for PINN models. Since many constructions in differential geometry may be framed as minimisation of a differential functional, these functionals can be coded as loss functions to align the AI loss-minimisation goal with that of solving the geometric problem. This contribution to the Recent Progress in Computational String Geometry workshop proceedings introduces the PINN architecture defining principles, motivates how they are well suited for problems in differential geometry, and demonstrates their use via summaries of three works at this intersection.

Problem

• PINNs
• differential geometry
• differential functional
• geometric problems
• computational geometry

Innovation

• Physics-Informed Neural Networks
• Differential Geometry
• Variational Problems
• Loss Function Design
• Computational String Geometry