Parameter Regrouping: Discovering Symmetries Within the Data

Many neural architectures gain efficiency by encoding known symmetries as fixed weight-sharing patterns. We study whether a useful sharing pattern can instead be learned from task gradients. We introduce parameter regrouping: connections share a small set of values, a group is split when its connection-level gradients disagree, and groups are merged when their values and mean gradients are similar. We study the mechanism through Shared Architecture Inference (SHAI), a proof-of-concept model for same-dimensional next-state prediction trained with layer-local losses during discovery. Across five dynamical benchmarks, SHAI obtains lower median test error than an MLP on four tasks while using $7$--$138\times$ fewer parameters, and lower error than a hand-designed convolution on four. Controlled teachers show value-faithful recovery when reusable sharing exists, a dense negative control incurs substantially higher error, and Lenia yields a ring-like kernel at the correct radius. The method remains limited to same-dimensional dynamics, over-splits, and scales quadratically with state dimension.