UniPhy: Unifying Riemannian-Clifford Geometry and Biorthogonal Dynamics for Planetary-Scale Continuous Weather Modeling

TL;DR

UniPhy tackles the mismatch between discrete-time data-driven weather models and the atmosphere's continuous, open, multi-scale dynamics. It integrates a Riemannian-Clifford geometric encoding, a non-Hermitian biorthogonal spectral propagator, a global flux memory tracker, and a log-linear parallel integration engine to form a physically principled, scalable continuous-time solver. The approach demonstrates non-normal transient growth, teleconnection memory, and zero-shot temporal generalization on ERA5 data, with a two-stage Thermodynamic Alignment training strategy to balance short-term accuracy and long-term stability. This framework holds promise for resolution-independent, physically consistent global weather modeling and efficient parallel inference on planetary scales.

Abstract

While data-driven weather models have achieved remarkable deterministic accuracy, they fundamentally rely on discrete-time mappings and closed-system assumptions, failing to capture the multi-scale continuous dynamics and thermodynamic openness of the atmosphere. To address these limitations, we propose UniPhy, a continuous-time non-Hermitian neural stochastic partial differential equation (SPDE) solver. Geometrically, we employ Riemannian-Clifford gauge transformations to flatten planetary heterogeneity, enabling globally consistent operations. Dynamically, we construct non-Hermitian biorthogonal spectral operators integrated with a global flux tracker to capture transient energy growth and open-system exchange. Computationally, by identifying the algebraic associativity of the analytic solution, we reformulate adaptive physical integration as a parallel prefix-sum problem, achieving log-linear sequence parallelism. UniPhy establishes a physically complete foundation model architecture that unifies geometric adaptivity, thermodynamic consistency, and computational efficiency. Our code is available at <https://github.com/yrqUni/UniPhy>.

Figures (7)

• Figure 1: The UniPhy Architecture. The framework is structured into four functional modules: (I) Geometric Encoding: The Riemannian-Clifford Encoder ($\mathcal{E}_\phi$) flattens the heterogeneous Earth manifold into latent states. (II) Adaptive Time Inputs: The model accepts variable time steps (e.g., 3h, 6h, 12h) to support multi-scale integration. (III) Semigroup Operator Generator: A continuous-time physics engine models state propagation via decay operators $\lambda$ and forcing terms $f(t)$, ensuring continuous consistency. (IV) Parallel Prefix-Scan Integration: By reformulating the recurrent evolution $h_t = A h_{t-1} + B u_t$ as an associative scan operation, the $O(\log T)$ engine parallelizes the temporal integration. Additionally, a Flux Tracker (bottom) with recurrent gating maintains long-term memory and global conservation across the sequence.
• Figure 2: Mechanistic Verification of Geometric Perception. (A) The spatial distribution of learned position embeddings $\sigma_\phi(x)$ adapts to the spherical topology and spontaneously highlights geographic features such as coastlines and the Tibetan Plateau. (B) The learned gauge weights significantly correlate with the analytical metric distortion and effectively learn a metric compensation mechanism.
• Figure 3: Evidence of Non-Normal Amplification. (A) The eigenspectrum of the learned operator is confined to the stable region where $\text{Re}(\lambda) < 0$. (B) The energy evolution reveals a massive transient spike shown in red which is characteristic of non-normal fluid instabilities such as storm formation. This contrasts with the monotonic decay of the normal baseline shown in black dashed lines.
• Figure 4: Emergence of Memory Hierarchy. The distribution of learned timescales shows a heavy tail. While 97.5% of modes focus on fast weather dynamics of less than 5 days, a critical 1.6% of modes capture long-term teleconnections exceeding 20 days. This validates the efficacy of the Global Flux Tracker.
• Figure 5: Qualitative Comparison of Zero-Shot Temporal Generalization on Mean Sea Level Pressure (MSLP). Rows display Ground Truth, Pre-trained, and Fine-tuned models. Columns show snapshots at $t=1h$, $6h$, and $12h$ derived from a fine-grained integration step of $\Delta t=1h$. The Fine-tuned model (bottom row) preserves sharper structural details and physical consistency at the unseen $\Delta t=1h$ step compared to the Pre-trained model (middle row).

Theorems & Definitions (4)

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