Hamiltonian neural networks provide a principled framework for learning conservative dynamics. A scalar Hamiltonian is parameterized by a neural network and the vector field is derived using Hamilton's equations. However, preserving Hamiltonian structure does not guarantee that the energy function itself is represented accurately. When modeled by a conventional multilayer perceptron, the Hamiltonian inherits the spectral bias of coordinate-based neural representations, favoring smooth over fine-scale structure. These fine-scale errors can be amplified by differentiation when computing the dynamics. To address this limitation, we introduce Local Multiscale Hamiltonian Neural Network (LM-HNN), which restructures the Hamiltonian according to its physical interaction structure. LM-HNN decomposes the energy into kinetic and potential components and represents them as sums of shared local functions over multiple interaction neighborhoods. This replaces a single high-dimensional approximation problem with lower-dimensional, reusable interaction laws, making fine-scale energy structure easier to resolve while preserving Hamiltonian dynamics. Across low-dimensional, interacting, and fast--slow systems, LM-HNN consistently achieves more accurate Hamiltonian derivatives and long-horizon rollouts than competing structure-preserving models, while substantially extending the effective spectral range of the learned energy. The same structured representation also generalizes more reliably to energy levels not observed during training. These results show that accuracy depends on the Hamiltonian representation as well as its structure, and that incorporating locality, decomposition, and shared interaction laws provides a simple and effective route to more accurate and generalizable Hamiltonian learning.
Spherical-harmonic neural operators provide an efficient representation for global forecasting, but their discretized pointwise nonlinearities can break rotational equivariance through aliasing. Clebsch-Gordan (CG) tensor products provide exactly equivariant interactions between spherical-harmonic representations, but their computational cost grows rapidly with spectral bandwidth and has limited their use at forecasting scale. We introduce LCG-SFNO, which combines low-rank CG interactions within a controlled low-degree band with coefficient-space residual propagation and an exactly projected quadratic nonlinearity, preserving SO(3) equivariance throughout the learned operator. To make the CG interaction practical, we further develop SSHCG, a sparse GPU implementation with a factorized alternative for wider coupled bands. On spherical shallow-water dynamics and the ERA5-based WeatherBench benchmark, LCG-SFNO reduces rotational error to numerical precision (more than three orders of magnitude below SFNO on shallow water and five orders of magnitude below it on WeatherBench), while matching SFNO on shallow-water forecasting and substantially improving forecast skill relative to SFNO on WeatherBench. Matched ablations show that CG coupling accelerates convergence, while SSHCG keeps the coupling to about 10 percent of a training step at the operating bandwidth, making exact equivariant coupling practical at forecasting scale.