PINNs in More General Geometry

TL;DR

This paper discusses how Physics-Informed Neural Networks (PINNs) can be adapted to solve complex problems in differential geometry by framing geometric conditions as differential functionals within neural loss functions.

Contribution

It introduces the principles of PINN architectures tailored for differential geometry and demonstrates their application through summaries of three relevant works.

Findings

PINNs can encode differential geometric problems as loss functions.

PINNs are well suited for solving problems in differential geometry.

The paper showcases three applications of PINNs in geometric contexts.

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.