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This paper explores the application of Physics-Informed Neural Networks (PINNs) to problems in differential geometry by framing geometric constructions as the minimization of differential functionals, which are then encoded as loss functions. It argues that PINNs are well-suited for differential geometry problems because their loss-minimization aligns with solving geometric problems. The paper summarizes three works demonstrating the use of PINNs in computational string geometry.
PINNs offer a promising new approach to solving complex problems in differential geometry by directly minimizing differential functionals.
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.
