Binghang Lu

Binghang Lu

Graduate Researcher, Purdue ECE — LLMs · Continual Learning · AI4S

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

Published in arXiv preprint, 2026

PINNs typically enforce boundary and physical constraints via soft penalty terms, which trade accuracy for trainability. AdamFLIP treats constraint satisfaction as a control problem: at each step it applies a feedback-linearization correction on top of Adam’s adaptive momentum, projecting the update onto the hard-constraint manifold. The resulting optimizer matches Adam’s convergence rate while delivering constraint residuals orders of magnitude smaller than soft-penalty baselines.

Algorithm 1: AdamFLIP. Computes a constraint multiplier λ_t from the constraint Jacobian, projects the raw gradient onto the constraint tangent, updates Adam's first and second moments, bias-corrects them, and applies the Adam step.
Algorithm 1. The AdamFLIP update. A closed-form feedback-linearization multiplier λ_t projects the raw gradient onto the constraint tangent before standard Adam momentum and adaptive-step bookkeeping, so the hard constraint h(θ) = 0 is enforced step-by-step rather than via a soft penalty.

Recommended citation: Lu, B., Zhang, R., Mou, C., Li, N., & Lin, G. (2026). "AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training." arXiv:2605.08408.
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