RAMPing Up PINN Training Signals via Parameter-Sensitivity-Guided Collocation Refinement

Collocation point selection plays a crucial role in training physics-informed neural networks (PINNs), as physics constraints are enforced only at sampled locations. Most existing adaptive sampling methods prioritize points with large constraint violations, typically measured by PDE residual magnitude, thereby implicitly treating residual magnitude as a sufficient pointwise indicator of training utility. We revisit adaptive collocation from a parameter-space perspective and introduce *pointwise parameter sensitivity*, defined as the norm of the gradient of the pointwise PDE loss with respect to the trainable parameters. We show that sampling in proportion to this quantity directly amplifies the induced parameter-space training signal, up to a directional coherence factor. Building on this criterion, we propose **R**adius-bounded **A**scent for **M**aximizing **P**arameter-gradient norm (**RAMP**), a parameter-sensitivity-guided collocation refinement strategy that adapts sampled points to the current parameter state. Given candidates drawn from a prescribed base sampler, RAMP independently refines each point by projected unit-gradient ascent on pointwise parameter sensitivity within a bounded radius. This local refinement preserves the spatial coverage induced by the base sampler while steering collocation points toward regions that provide stronger parameter-space signals. We further analyze magnitude-based sampling criteria and show that they can be viewed as residual-only approximations of parameter sensitivity, which neglect the local parameter responsiveness of the PDE residual. Across benchmark PDEs and multiple PINN optimizers, RAMP achieves consistently larger aggregate gains over the base sampler than recent adaptive sampling methods, while retaining parallelizable point updates.