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Climate Physics Dynamic Matching
Gurjeet Sangra Singh, Frantzeska Lavda, Alexandros Kalousis
TL;DR
ClimPhyDM targets weather forecasting models that either neglect atmospheric physics or require incomplete PDE specifications and solver-based training. It combines an advection prior with variational dynamics matching and neural components, then directly regresses finite-difference targets without solver backpropagation. On ERA5, it outperforms ClimODE and deterministic GB-DM at hourly and monthly resolutions, with growing margins at longer horizons.
Problem
Existing data-driven weather models can neglect atmospheric transport, while solver-based grey-box models incur integration costs and deterministic fields cannot represent multiple plausible unresolved evolutions.
Method
ClimPhyDM combines an advection-type physics prior, neural velocity and source terms, and a variational latent trained by directly matching physics-composed fields to finite-difference targets.
Results
On ERA5, ClimPhyDM outperforms ClimODE and its deterministic ablation at hourly and monthly resolutions, with growing margins at longer horizons.
Takeaways & Limitations
The framework provides a simulation-free grey-box approach with improved long-horizon forecasting behaviour and avoids solver backpropagation during training.
Abstract
from arXiv · showhide
Deep generative models such as flow matching and diffusion models have shown potential for learning complex dynamical systems, but typically act as black boxes that neglect underlying physical structure, while physics-based models governed by partial differential equations are often incomplete due to missing source terms, or uncertain parametrisations. We present Climate Physics Dynamic Matching (ClimPhyDM), a variational simulation-free dynamics informed framework for weather forecasting that combines an advection-type physics prior with data-driven components in a variational framework. % to capture the stochasticity and multi-modality of unresolved atmospheric dynamics. On the ERA5 benchmark at hourly (42-hour) and monthly (5-month) resolutions, ClimPhyDM outperforms ClimODE, and GB-DM, keeping the lower error at extended horizon, indicating improved temporal stability and resistance to error accumulation, while its simulation-free paradigm also enables training on a single modest 12 GB consumer GPU.
1 Introduction
ClimPhyDM addresses limitations of black-box and solver-based grey-box weather models by combining atmospheric physics with data-driven learning. Its variational, simulation-free formulation represents multiple plausible unresolved dynamics without backpropagating through solver steps.
- Black-box forecasting models learn dynamics purely from data, which can produce physically implausible predictions, poor generalisation, and limited interpretability.
- Solver-based grey-box training requires forward integration and backpropagation through every solver step, increasing memory and training time and risking adjoint instability.The memory cost scales linearly with the number of integration steps.
- Deterministic learned vector fields can average multiple plausible evolutions admitted by incomplete physics, blurring unresolved atmospheric dynamics.
- ClimPhyDM combines atmospheric advection dynamics with a neural corrector and uses finite-difference target velocities for single-forward-pass dynamics matching.
- A variational latent inferred from recent state histories captures stochasticity and multi-modality in unresolved dynamics while avoiding solver backpropagation.
2 Method
ClimPhyDM builds a grey-box atmospheric dynamics model from an advection-type PDE prior, neural velocity and source terms, and a variational latent. It trains by directly matching the physics-composed vector field to finite-difference targets rather than simulating trajectories during optimization.
- Climate Physics Dynamics: The model represents atmospheric states on a latitude-longitude grid and uses an advection-type PDE as its physics prior for continuous-time dynamics.
- Climate Physics Dynamics: Neural velocity and source terms condition on the state, spatial gradients, stochastic latent, and fixed spatio-temporal embeddings.The source term models non-advective processes including diabatic heating, radiative forcing, and subgrid turbulence.
- Climate Physics Dynamics: The physics-composed vector field applies advection and compression through the advection operator, then adds a flexible source term.This construction preserves transport and compression structure while retaining flexibility for non-advective dynamics.
- Simulation-Free Variational Dynamics Matching: For consecutive states, the method constructs an interpolation path and uses its finite-difference derivative as a supervised target for the vector field.
- Simulation-Free Variational Dynamics Matching: A variational posterior conditioned on recent state history disambiguates plausible velocity modes, with the ELBO combining expected matching error and a KL regularizer.The interpolation time is sampled uniformly between consecutive state times.
3 Experiments
ClimPhyDM is evaluated on ERA5 at hourly and monthly resolutions against ClimODE, GB-DM, and persistence, with performance assessed through forecast horizons.
- ERA5 experiments use five meteorological variables at 5.625° spatial resolution and 6-hour intervals, with training on 2006–2015, validation on 2016, and testing on 2017–2018.
- The evaluation covers hourly forecasts up to 42 hours and monthly forecasts up to 5 months, comparing ClimPhyDM with ClimODE, GB-DM, and persistence.GB-DM isolates the variational latent contribution, while ClimODE isolates simulation-free training.
- ClimPhyDM achieves lower RMSE and higher ACC than ClimODE and GB-DM across the 42-hour hourly horizon, retaining temporal stability as lead time increases.Residual maps also show lighter error structures across all five variables, particularly over oceanic regions.
- At the 5-month horizon, the performance gap to ClimODE widens after the third month, indicating slower error accumulation and more consistent behaviour at extended lead times.
4 Conclusion
ClimPhyDM combines an advection-type physics prior with variational dynamics matching in a simulation-free grey-box framework. On ERA5, it outperforms ClimODE and deterministic GB-DM at hourly and monthly resolutions, with larger margins at longer horizons.
- ClimPhyDM combines an advection-type physics prior with variational dynamics matching for weather forecasting.
- On ERA5, ClimPhyDM outperforms ClimODE and deterministic GB-DM at hourly and monthly resolutions, with growing margins at longer horizons.
- Flow matching constructs interpolation paths between paired states and learns target velocities without solving the continuity equation through forward simulation.The objective uses a single forward pass per training step without numerical integration.
- Trajectory Flow Matching extends conditional flow matching to consecutive states in time-series trajectories, learning the velocity field governing temporal evolution.
- Grey-box dynamics compose a neural network with a physics operator, allowing learned inputs to modulate the operator rather than merely adding an output correction.This composition is described as enforcing physical constraints on the learned dynamics.
- ClimPhyDM avoids the O(L) memory and computational cost associated with solver-based grey-box training and backpropagation through L integration steps.
A.3 ELBO Derivation
The ELBO derivation rewrites the conditional generative-model objective using a latent-variable likelihood and a variational posterior. Assuming a time- and data-independent prior yields an expected log-likelihood minus a KL divergence.
- The ELBO is used as the loss objective for training the conditional generative model.
- Jensen’s inequality lower-bounds the log marginal likelihood by integrating the joint model over the variational posterior.
- The derivation factors the joint distribution into a conditional likelihood and a latent prior before comparing it with the variational posterior.
- Assuming p_t^ϕ(z | u_k) = p(z), the ELBO becomes an expected conditional log-likelihood minus KL[q_ψ(z | u_{k-h:k}) || p(z)].
B.1 Evaluation metrics
The evaluation uses latitude-weighted RMSE and ACC after denormalizing predictions, with latitude weighting and empirical-mean anomalies defined for the metric calculations.
- Latitude-weighted RMSE and Anomaly Correlation Coefficient are computed after predictions are denormalized.
- The latitude weight is cos(h′), and anomaly fields subtract the empirical mean C before comparison.
B.2 Dataset details
The study uses a coarsened, six-hourly ERA5 dataset with five meteorological variables, divided into training, validation, and test periods.
- Dataset resolution: ERA5 data are regridded to 5.625° spatial resolution and sampled at 6-hour intervals.WeatherBench provides the preprocessing from the original 0.25° ERA5 data.
- Meteorological variables: The experiments model five variables: 2-metre temperature, atmospheric temperature, geopotential, and the 10-metre wind components.All variables are min–max normalized to [0, 1].
- Data splits: The dataset contains ten training years from 2006–2015, one validation year in 2016, and two test years from 2017–2018.The full set of ERA5 variables is summarized in Table A1.
B.3 Variational model and training
The model uses residual convolutional networks to represent physics-informed velocity and source terms, and trains them with a variational ELBO and smoothed stochastic matching targets.
- Model architecture: Two residual convolutional networks instantiate the velocity field vϕ and source sϕ, sharing state, gradient, time-embedding, and static-field inputs.The velocity network is deeper than the source network, while spherical-grid padding and self-attention support spatial processing.
- Optimization: Training optimizes the ELBO with AdamW, cosine learning-rate decay, gradient clipping, and linear KL annealing.The learning rate decays from 5 × 10^-4 to 10^-5, while the KL weight increases from zero to 0.1.
- Matching target: A stochastic interpolant anneals its noise level from 10^-2 to 10^-3, smoothing the velocity-field matching target during training.The procedure is described as a smoothing regulariser on the velocity field.
- Dataset variables: Table A1 organizes the ERA5 variables by static, single-level, and atmospheric categories.Static variables are time-independent; single-level and atmospheric variables represent different meteorological data types.
- Evaluation: The reported 42-hour forecast compares predicted and true values through their error relative to ClimODE.This comparison is presented for the ERA5 dataset.
C Related Work
Related work spans data-driven forecasting, physics-informed learning, grey-box dynamical systems, and simulation-free flow-matching methods.
- Data-driven weather forecasting: Pangu-Weather, GraphCast, and FourCastNet use transformer, graph, and Fourier-neural-operator approaches for data-driven weather forecasting.GenCast applies conditional diffusion models for ensemble weather prediction.
- Forecast example: Figure A5 presents an example of ClimPhyDM climate forecasting over a 42-hour horizon.The figure is identified as an example forecast rather than a general method description.
- Physics-informed neural networks: Physics-informed methods such as PINNs and Fourier Neural Operators incorporate physical knowledge through constraints, penalties, or structural operators.These methods typically require governing equations in full rather than incomplete physics models.
- Grey-box dynamical systems: Grey-box methods combine incomplete physics with neural networks, but the cited approaches rely on Neural ODE solvers during training.This includes PhysVAE and other hybrid-learning methods for complex dynamics.
- Flow matching and trajectory learning: Conditional and Trajectory Flow Matching provide simulation-free training for generative and time-series models, while Variational Rectified Flow Matching introduces latent variables for multi-modal velocity fields.ClimPhyDM adapts trajectory flow matching to a grey-box setting with compositional physics embedding.