Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization

We study how neural emulators of partial differential equation solution operators internalize physical symmetries by introducing an influence-based diagnostic that measures the propagation of parameter updates between symmetry-related states, defined as the metric-weighted overlap of loss gradients evaluated along group orbits. This quantity probes the local geometry of the learned loss landscape and goes beyond forward-pass equivariance tests by directly assessing whether learning dynamics couple physically equivalent configurations. Applying our diagnostic to autoregressive fluid flow emulators, we show that orbit-wise gradient coherence provides the mechanism for learning to generalize over symmetry transformations and indicates when training selects a symmetry compatible basin. The result is a novel technique for evaluating if surrogate models have internalized symmetry properties of the known solution operator.

Symmetry constrained neural networks for detection and localization of damage in metal plates

The present paper is concerned with deep learning techniques applied to detection and localization of damage in a thin aluminum plate. We used data generated on a tabletop apparatus by mounting to the plate four piezoelectric transducers, each of which took turn to generate a Lamb wave that then traversed the region of interest before being received by the remaining three sensors. On training a neural network to analyze time-series data of the material response, which displayed damage-reflective features whenever the plate guided waves interacted with a contact load, we achieved a model that detected with greater than 99% accuracy in addition to a model that localized with $3.14 \pm 0.21$ mm mean distance error and captured more than 60% of test examples within the diffraction limit. For each task, the best-performing model was designed according to the inductive bias that our transducers were both similar and arranged in a square pattern on a nearly uniform plate.