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

2026-05-08

AI Summary

This work addresses the ill-posed optimization, sensitivity to loss weights, and insufficient constraint satisfaction commonly observed in physics-informed neural networks (PINNs) due to soft constraints. The authors reformulate PINN training as an equality-constrained optimization problem and, for the first time, introduce concepts from feedback linearization control to explicitly compute Lagrange multipliers via a dynamical system. By integrating Adam’s adaptive first- and second-moment estimation within this hard-constrained framework, the method achieves stable contraction dynamics and efficient gradient-based optimization. Evaluated across multiple forward and inverse partial differential equation benchmarks, the proposed approach significantly outperforms existing PINN variants, reducing the relative L2 error of predicted solutions by more than two-thirds on the Navier–Stokes equations.

Abstract

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.

Problem

• Physics-informed neural networks
• hard constraints
• constrained optimization
• PDEs
• loss weighting

Innovation

• Physics-informed neural networks
• Hard constraints
• Feedback linearization
• Adaptive optimization
• Lagrange multipliers