Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains.

Abstract: We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided reasonable integration schemes are available. Applications to traveling waves, breathers and solitons, as well as numerical experiments confirming the validity of the approximation are also presented as well.

Posted on Jan 21, 2025 in Differential Equations, Seminars