Organizations: Department of Mathematical Sciences / Montana State University, Bozeman, Montana, US · Department of Mathematics / Imam Khomeini International University, Qazvin, Iran · Department of Electrical and Computer Engineering, Montana State University, Bozeman, Montana, US · Department of Mechanical and Industrial Engineering, Montana State University, Bozeman, Montana, US
Short-horizon forecasts of atmospheric temperature are needed to support climate-aware digital-twin systems, but such forecasts must be produced where thermal observations are incomplete. This study evaluates a physics-informed neural network for potential-temperature forecasting, constrained by a pressure-coordinate thermodynamic advection-source equation and a diabatic-source closure fit from the preceding 12-hour period and frozen before future-time training. Using hourly ERA5 reanalysis at three pressure levels, the model is evaluated as a conditional hindcast at lead times of one, two and three hours against persistence, local-trend, and two matched neural-network baselines, one of which receives the same future meteorological forcing as the PINN, helping distinguish the physical constraint from access to future forcing. In an Oklahoma development case, mean RMSE improvement over the strongest baseline grew from 8.1% at one hour to 23.8% at three hours; under an observation-density sweep down to 5% of candidate locations, this 3-hour advantage remained 14.6--16.9%, with no evidence that lower density improves performance. Under a fixed protocol transferred to an Alabama heat event with three virtual-observation layouts, three-hour improvement ranged 19.7-24.4% with consistent origin-level wins. A parallel Montana stress test, in which fixed pressure levels intersected complex terrain, produced a three-hour degradation of roughly 17.5%, identifying a terrain-related applicability limit of the formulation. Together, these results indicate that the physics constraint's benefit grows with forecast horizon, persists under severe observation sparsity, and transfers across regions, but is bounded by the validity of a fixed vertical-coordinate representation over complex terrain, evidence relevant to physics-constrained components of climate-aware forecasting and digital-twin systems.
Figures & tables
Region / experiment
Inner evaluation domain
Forecast origins (UTC)
Temperature setting
Experimental role
Oklahoma development
33.5∘ – 37.0∘ N, 100.0∘ – 94.5∘ W
Aug. 15 12:00, 18:00; Aug. 16 12:00, 18:00
Dense
Model development
Oklahoma chronological replication
Same
Aug. 19 12:00, 18:00; Aug. 20 12:00, 18:00
Dense
Later-period replication under the fixed protocol
Oklahoma density study
Same
Same four origins
25%, 10%, 5%; dense reference
Observation-density × forecast-horizon analysis
Alabama
31.0∘ – 34.5∘ N, 88.5∘ – 85.0∘ W
Jul. 27 12:00, 18:00; Jul. 28 12:00, 18:00
10%
Fixed-protocol cross-region evaluation and layout robustness
Montana
45.0∘ – 47.5∘ N, 109.0∘ – 104.5∘ W
Jul. 25 12:00, 18:00; Jul. 26 12:00, 18:00
Nominal 10%
Terrain/applicability stress test
Table 1: Summary of the experimental design across study regions.
Experiment
Mean PINN RMSE (K)
Mean strongest- baseline RMSE (K)
Mean RMSE improvement (%)
Overall PINN win rate (%)
+3 -h PINN win rate (%)
Oklahoma development
0.526
0.636
17.3
70.8
62.5
Oklahoma chronological replication
0.641
0.781
18.0
75.0
75.0
Table 2: Oklahoma development and chronological-replication performance under dense temperature observations.
Temperature-observation coverage
+1 -h improvement (%)
+2 -h improvement (%)
+3 -h improvement (%)
Overall mean improvement (%)
Overall PINN win rate (%)
+3 -h PINN win rate (%)
Dense (100%)
8.1
14.7
23.8
18.0
75.0
75.0
25%
2.6
5.5
16.9
10.1
75.0
87.5
10%
1.8
6.3
15.1
9.3
62.5
87.5
5%
1.4
9.1
14.6
9.8
70.8
87.5
Table 3: Oklahoma PINN performance by temperature-observation coverage and forecast lead time.
Figure 1: Effect of temperature-observation coverage and forecast horizon on Oklahoma forecast performance.
Forecast lead
PINN RMSE (K)
Strongest-baseline RMSE (K)
RMSE improvement (%)
PINN win rate (%)
+1 h
0.373
0.391
4.6
37.5
+2 h
0.573
0.646
11.3
75.0
+3 h
0.691
0.860
19.7
75.0
Table 4: Alabama cross-region evaluation and robustness to sparse-observation layout.
Figure 2: Spatial comparison for a representative +3 -h Alabama forecast at 850 hPa under 10% temperature-observation coverage. The six panels show: (a) ERA5 temperature field and virtual observation locations, (b) PINN forecast, (c) persistence forecast, (d) PINN absolute error, (e) persistence absolute error, and (f) three-dimensional PINN temperature surface. Panels (a)–(c) use a common temperature scale, while panels (d) and (e) use a common absolute-error scale.
Forecast lead
Mean PINN RMSE (K)
Mean strongest- baseline RMSE (K)
RMSE improvement (%)
+1 h
3.201
2.954
−8.4
+2 h
3.363
2.951
−14.0
+3 h
3.458
2.942
−17.5
Table 5: Montana frozen stress-test performance by forecast lead.
Figure 3: Terrain applicability and forecast performance in the Montana stress test.
This study introduces enhancements to physics-constrained neural networks (PCNNs) that improve the accuracy and stability of hybrid short-term weather forecasting models. Building on the WeatherGFT architecture, three innovations are proposed. First, an upgraded numerical solver, combining a fifth-order weighted essentially non-oscillatory scheme (WENO-5), a beta-plane approximation, and subgrid-scale viscosity, permits a fourfold increase in the integration time step to 1200 s while reducing the daily mean squared error by up to 26%. Second, a unified autoregressive hybrid block replaces the original chain of 24 specialised modules, eliminating overfitting to specific lead times. Third, the physical core is integrated with two state-of-the-art neural backbones, resulting in PI-PredFormer and PI-IAM4VP. Evaluation on the WeatherBench South Pacific subset from 2000 to 2004 shows that these hybrids reduce root mean squared error at 1-12 h lead times by 8-22% compared to purely neural counterparts, while better preserving physical consistency. These results demonstrate that incremental refinement of hybrid components offers a practical route toward more accurate and efficient short-range weather forecasting.
Egor Bugaev, Fedor Buzaev, Dmitry Efremenko +2
Faculty of Computer Science Higher School of Economics Moscow, Russia
Operational weather prediction has long relied on physics-based numerical weather prediction (NWP), whose accuracy comes at the cost of substantial compute and complex simulation workflows. Recent transformer-based forecasters offer efficient data-driven alternatives, however transformers are physics-agnostic models. Additionally, standard transformer encoders evolve representations through discrete layer updates that may be less suited to modeling smooth latent dynamics. In this work, we propose a continuous-depth transformer encoder for weather forecasting that integrates Neural Ordinary Differential Equation (Neural ODE) dynamics within each encoder block. Specifically, we replace discrete residual updates with ODE-based updates solved using adaptive numerical integration. We also introduce a two-branch attention module that combines conventional patch-wise self-attention with an auxiliary branch that applies a derivative operator to attention logits, providing an additional change-sensitive interaction signal. To further align forecasts with governing principles, we propose a customized physics-informed training objective that enforces physical consistency as a soft constraint. We evaluate the proposed method against a standard discrete transformer baseline and an existing continuous-time Neural ODE forecasting variant, demonstrating the importance of PINN-Cast in short term weather forecasting.
Hira Saleem, Flora Salim, Cormac Purcell
University of New South Wales, Sydney, NSW, Australia
Time series forecasting has seen signicant advancements with the emergence of new deep learning models. However, forecasting time series in applications involving physical processes remains a major challenge. Despite the apparition of Physics Informed Neural Networks (PINN), recent models do not estimate unobservable intermediate physical variables, which are important for domain experts to understand the target behavior. To this end, we propose a Physics Informed Recurrent Neural Network (PIRNN) which predicts, along the target, unobservable variables on both historic data and forecast target. This approach enhances the model robustness and results interpretation using domain knowledge. Our method is easily adaptable to any physical model using several equations, each having its own set of unobservable variables, to describe it-self. As a case study, we incorporate physical equations used for groundwater levels predictions by the physical model called Gardenia. This model uses transfers equations between reservoirs, optimized with data assimilation, to simulate the evolution of groundwater levels. Evaluation includes several well known neural network models and the Gardenia model compared on twelve real world datasets. In addition, we study the impact of each component through an ablation study. Our model outperforms other models on ve out of the twelve datasets and our ablation study underlines the importance of having a physical background in our time series forecasting task. Finally, the coherence of the physical variables predicted by our neural network is assessed by a domain expert.
Etienne Lehembre, Pascal Audigane, Vincent Nguyen +2