Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEs

What the math gives to ML

The paper's transferable asset is a formulation-independent primal–physical-adjoint error identity: the error in a prescribed quantity of interest is bounded by the product of primal and adjoint residual errors, without relying on Galerkin orthogonality. This suggests training physics-informed neural networks with a jointly learned physical adjoint rather than using only the primal PDE residual or a least-squares-induced dual. The exact elementwise residual decomposition can be converted into an adaptive collocation rule that concentrates computation where both primal and adjoint errors jointly affect the target output. The strongest initial test is a dual-weighted PINN for a mixed-boundary elliptic PDE, evaluated on QoI accuracy at fixed collocation and optimization budgets.