# Physically Typed and Geometry-Aware Representations for Earth Foundation Models

arXiv: https://arxiv.org/abs/2609.13868 · HTML: https://mlsanity.com/paper/2609.13868/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 45 references that are in the map.
- First posted on arXiv 12 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SatMAE: Pre-training Transformers for Temporal and Multi-Spectral Satellite Imagery](https://mlsanity.com/paper/2207.08051/) (2022)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [WeatherBench 2: A benchmark for the next generation of data-driven global weather models](https://mlsanity.com/paper/2308.15560/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [CROMA: Remote Sensing Representations with Contrastive Radar-Optical Masked Autoencoders](https://mlsanity.com/paper/2311.00566/) (2023)
- [Foundation Models for Generalist Geospatial Artificial Intelligence](https://mlsanity.com/paper/2310.18660/) (2023)
- [Prithvi-EO-2.0: A Versatile Multi-Temporal Foundation Model for Earth Observation Applications](https://mlsanity.com/paper/2412.02732/) (2024)
- [AlphaEarth Foundations: An embedding field model for accurate and efficient global mapping from sparse label data](https://mlsanity.com/paper/2507.22291/) (2025)
- [Neural Plasticity-Inspired Multimodal Foundation Model for Earth Observation](https://mlsanity.com/paper/2403.15356/) (2024)
- [TerraMind: Large-Scale Generative Multimodality for Earth Observation](https://mlsanity.com/paper/2504.11171/) (2025)
- [Galileo: Learning Global & Local Features of Many Remote Sensing Modalities](https://mlsanity.com/paper/2502.09356/) (2025)
- [OlmoEarth: Stable Latent Image Modeling for Multimodal Earth Observation](https://mlsanity.com/paper/2511.13655/) (2025)
- [TESSERA v2: Scaling Pixel-wise Earth Foundation Models](https://mlsanity.com/paper/2607.03949/) (2026)
- [HDNet: Physics-Inspired Neural Network for Flow Estimation based on Helmholtz Decomposition](https://mlsanity.com/paper/2406.08570/) (2024)
- [SwissCrop25: A National Multi-Year Benchmark for Operational Crop Mapping](https://mlsanity.com/paper/2608.09497/) (2026)
- [SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres](https://mlsanity.com/paper/2503.09456/) (2025)

## Abstract (arXiv)

Earth-observation (EO) foundation models have become exceptionally effective at learning se mantic, high-dimensional geospatial embeddings, while modern weather and climate models have demonstrated that Earth-specific geometry, spherical operators, meshes, and hybrid physical solvers can materially improve prediction. Yet these two advances are not equivalent. A conventional latent embedding has no inherent physical transformation law, whereas scalar fields, tangent polar-vector fields, axial/pseudovector quantities, covectors, and higher-order tensors transform differently under rotations, reflections, and changes of local coordinate frame. This proposal asks whether a general purpose Earth foundation model should preserve those distinctions explicitly, or whether standard embeddings plus augmentation already learn everything that matters. The central contribution is therefore not a more complicated architecture by assumption, but a staged falsification program. A compute-conscious ERA5 dry run first compares conventional, augmentation-matched, typed equivariant, and Hodge/Helmholtz variants under spatial, temporal, orientation, and low-data shifts. Only if explicit geometric typing yields reproducible improvements does the program advance toward a multimodal Earth foundation model in which semantic embeddings coexist with physically typed fields. The proposed gap is narrower and more defensible than claiming that current models ignore geometry entirely: several systems already respect spherical domain geometry, and emerging work explicitly learns scalar/vector fields on spheres. The unresolved question is whether foundation-scale, multimodal, parity-aware field typing produces practical gains beyond those existing approaches.