Distance-Residual Physics-Informed Neural Networks: A Deep Learning Framework for Differential and Partial Differential Inclusions

This paper introduces Distance-Residual Physics-Informed Neural Networks (DR-PINNs) for learning solutions governed by ordinary and partial differential inclusions. The framework targets governing laws in which a differential operator is constrained to lie within a set-valued map, using the squared distance to that map in place of the classical pointwise PDE or ODE residual.