LM-PIELM: A Lagrange-Multiplier Framework for Hard Constraint Enforcement in Physics-Informed Extreme Learning Machines for PDE-Governed Forward and Inverse Problems

Physics-Informed Extreme Learning Machines (PIELMs) solve PDEs as linear least-squares problems by fixing the feature layer and learning only the output weights, but boundary conditions are usually imposed through soft penalties with problem-dependent weights. We propose LM-PIELM, a Lagrange-multiplier formulation that imposes boundary conditions as algebraic equality constraints using the linear-in-parameters structure of PIELMs. The resulting constrained least-squares problem leads to a linear saddle-point system with no boundary penalty tuning, no iterative dual updates, and no problem-specific analytical constrained ansatz. An equivalent augmented-system implementation avoids explicit normal-equation Gram matrices and improves robustness in ill-conditioned regimes. Since the constraint block uses only boundary operator evaluations, the same formulation applies to mixed boundary conditions, irregular geometries, and inverse problems including parameter identification and source recovery from noisy data. Across forward and inverse benchmarks, the augmented-system form preserves small boundary residuals, improves solution accuracy in ill-conditioned regimes, and keeps boundary admissibility essentially decoupled from sensor count, noise level, and reconstruction quality.