AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training

Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems governed by partial differential equations (PDEs), but standard PINN training typically relies on soft penalty formulations that combine PDE residuals, data mismatch, and initial/boundary conditions using manually chosen weights. This often leads to ill-conditioning, sensitivity to loss weights, and poor constraint satisfaction. In this work, we reformulate PINN training as an equality-constrained optimization problem and propose a novel Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN (AdamFLIP). The key idea is to view the constraint residuals as the output of a controlled dynamical system and to compute the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam-style first- and second-moment adaptation to the resulting feedback-linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural-network optimization. We test AdamFLIP on a range of benchmark forward and inverse PDE problem, and it consistently outperforms both the standard soft-constrained PINN and state-of-the-art constrained optimizers. Specifically, on the Navier--Stokes equations benchmark, AdamFLIP \textbf{reduces relative $L_2$ error by more than two thirds} for the predicted solution compared to the next best method. Our AdamFLIP framework provides an effective and computationally scalable hard constraint optimization method for PINN training.