PDE-State Conditioning for Learned Optimization Using Physics-Informed Neural Networks

Physics-informed neural network (PINN) optimization depends on global properties of the underlying PDE and how the physics residual evolves across the domain. Standard optimizers such as Adam use fixed gradient-based statistics, while learned PINN optimizers replace the fixed rule with a learned update built from local parameters and optimization-history features. However, none explicitly use PDE-level information. We study two coupled design questions: what PDE-level information should a learned optimizer observe, and how should that information influence its update? We introduce an 11-dimensional PDE state capturing operator structure, residual geometry, and short-horizon update response, and use it to condition a structured meta-learned optimizer through bounded feature-wise modulation. Repeated design studies identify the compact residual-geometry state G2 and bounded FiLM as the strongest overall choices among those tested. The resulting PDE-State-conditioned optimizer improves all evaluation metrics (relative-$L_2$, $L_1$, solution MSE, and physics MSE) over Adam and the unconditioned learned optimizer on Linear Advection, KdV, and Burgers.