Distance-Residual Physics-Informed Neural Networks: A Deep Learning Framework for Differential and Partial Differential Inclusions
This study addresses the challenge of solving differential inclusions where differential operators are constrained by set-valued mappings, a task for which conventional pointwise residual methods prove inadequate. To this end, we propose the DR-PINNs framework, which replaces traditional point residuals with differentiable distance residuals and computes the loss function via metric projections. By integrating convex quadratic programming with the chain rule, the approach effectively handles state-dependent constraints, supported by a theoretical consistency proof under continuous functionals. The proposed method achieves efficient approximation of solutions for both ordinary and partial differential inclusions. Benchmark evaluations demonstrate its superior accuracy and numerical stability, establishing DR-PINNs as a novel paradigm that combines rigorous theoretical guarantees with computational efficiency for solving differential inclusion problems.