DiffLiG: Diffusion-enhanced Liquid Graph with Attention Propagation for Grid-to-Station Precipitation Correction
Authors: Yuxiang Li
Yang Zhang
Guowen Li
Mengxuan Chen
Meng Jin
Fang Wang
Haohuan Fu
Juepeng Zheng
Venue: NeurIPS 2025 poster ()
Keywords: Precipitation Interpolation
Graph Neural Networks
Meteorological Post-processing
Primary area: machine_learning_for_sciences
TL;DR: We propose DiffLiG, a graph-based model that addresses station heterogeneity and oversmoothing via spatiotemporal encoding and diffusion-based ensemble refinement for accurate, structure-aware precipitation interpolation.

Abstract: Modern precipitation forecasting systems, including reanalysis datasets, numerical models, and AI-based approaches, typically produce coarse-resolution gridded outputs. The process of converting these outputs to station-level predictions often introduces substantial spatial biases relative to station-level observations, especially in complex terrains or under extreme conditions. These biases stem from two core challenges: (i) $\textbf{station-level heterogeneity}$, with site-specific temporal and spatial dynamics; and (ii) $\textbf{oversmoothing}$, which blurs fine-scale variability in graph-based models. To address these issues, we propose $\textbf{DiffLiG}$ ($\underline{Diff}$usion-enhanced $\underline{Li}$quid $\underline{G}$raph with Attention Propagation), a graph neural network designed for precise spatial correction from gridded forecasts to station observations. DiffLiG integrates a GeoLiquidNet that adapts temporal encoding via site-aware OU dynamics, a graph neural network with a dynamic edge modulator that learns spatially adaptive connectivity, and a Probabilistic Diffusion Selector that generates and refines ensemble forecasts to mitigate oversmoothing. Experiments across multiple datasets show that DiffLiG consistently outperforms other methods, delivering more accurate and robust corrections across diverse geographic and climatic settings. Moreover, it achieves notable gains on other key meteorological variables, underscoring its generalizability and practical utility.

DiffLiG: Diffusion-enhancee Liquid Graph with Attention Propagation for Grid-to-Station Precipitation Correction
Yuxiang Li, Yang Zhang', Guowen Li', Mengxuan Chen", Meng Jin", Fang Wang",
Haohuan Fu Juepeng Zheng! 13,1
Sun Yat-sen University "Tsinghua University
National Supercomputing Center in Shenzhen *Huawei Technologies Co., Ltd CCMA Earth System Modeling and Prediction Centre
Abstract
Modern precipitation forecasting systems, including reanalysis datasets, numerical models, and Al-based approaches, typically produce coarse-resolution gridded outputs. The process of converting these outputs to station-level predictions often introduces substantial spatial biases relative to station-level observations, especially in complex terrains under extreme conditions. These biases stem from two core challenges: (i) station-level heterogeneity, with site-specific temporal and spatial dynamics; and (ii) oversmoothing, which blurs fine-scale variability in graph- based models. To address these issues, we propose DiffLiG (Diffusion-enhanceel Liquid Graph with Attention Propagation), a graph neural network designed for precise spatial correction from gridded forecasts to station observations. DifILiG integrates GeoLiquidNet that adapts temporal encoding via site-aware OU dy- namics, a graph neural network with a dynamic edge modulator that learns spatially adaptive connectivity, and Probabilistic Diffusion Selector that generates and refines ensemble forecasts to mitigate oversmoothing. Experiments across multiple datasets show that DiffLiG consistently outperforms other methods, delivering more accurate and robust corrections across diverse geographic and climatic set- tings. Moreover, achieves notable gains other key meteorological variables, underscoring its generalizability and practical utility.
Introduction
Weather forecasting has gained increasing attention in both research and society, driven by rapid advances in forecast accuracy and temporal resolution. one hand, traditional Numerical Weather Prediction (NWP) systems, such the European Centre for Medium-Range Weather Forecasts (ECMWF)'s Integrated Forecasting System - High Resolution (IFS-HRES) and Integrated Forecasting System Ensemble (IFS-ENS, have achieved remarkable success in multiscale forecasts through high-resolution physical modeling, data assimilation, and ensemble strategies. On the other hand, the recent surge in Al-based weather models has introduced data-driven paradigm that bypasses explicit physical parameterizations 227
32 Sd] offering superior scalability and efficiency. For instance, FourCastNet (300 leverages Fourier neural operators for tempioltemporal coupling, Pangu-Wes Weather [5] employs 3D attention to improve mid-range accuracy, and GraphCast [22] utilizes spherical GNNs for 10-day forecasts. FuXi [10] demonstrates strong potential to extend the forecast horizons through hierarchical transformer framework tailored for long-range weather prediction.
Corresppnniing author: zhengjp80mail, www.l.yyau... adw..edu..u.........o edu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Despite these advances, both the NWP and Al models primarily operate on regular- regularrresolution grid oulputs, which are insufficient for real-world applications requiring high-resolution, station-level forecasts. In practice, operational users (such as urban planners, airport operators, or hydrological managers) require precise local forecasts at specific stations. However, model outputs often fail to generalize across diverse topographies or under sparse observational coverage, leading to spatial biases and degraded local accuracy. We provide a detailed empirical analysis of such interpolation emrors in the Appendix C
further concern arises from the evaluation process itself: most Al-based forecasting models are trained and validated on reanalysis data such as ERA5 [17], which integrates historical observations with simulated fields. As illustrated in Figure 6 this integration can introduce systematic biases, particularly in regions with sparse observations or complex terrain. While temporally consistent and spatially smooth, these products are not equivalent real-world station observations. As noted by Ramavajjala & Mitra 33 models like FourCastNet may outperform NWP systems in ERA5,, but fail retain such advantages when evaluated in ground-truth datasets (e.g.. NOAA'S MADIS), exposing considerable gap between simulation data and realism data in current Al weather systems.
Therefore, enable accurate and reliable station-level forecasting, it is crucial to move beyond uniform, coarse-to-fine interpolation strategies and develop models that can adapt to the heterogeneous nature of stations while preserving local spatial structures. This calls for dedicated interpolation framework that is both aaatinn-awar and uncertainty mnoerraint-innfommed,, capable of correcting the structural mismatch between gridded forecasts and real-world observations.
Station-level interpolation faces two fundamental chal- lenges: station heterogeneity and oversmoothing. The former refers to the diverse geographic, climatic, and ob- servational characteristics across stations. However, most existing interpolation methods adopt uniform architectures or fixed graph structures, without explicitly adapting the localized properties of individual stations, which often leads to suboptimal generalization and increased local er- rors, particularly in regions with complex terrain or sparse observations. Oversmoothing
Challenge Heterogeneity
Challenge Oversmoothing
station
c
Forecast Diffuser & Selector
interpolation is another widely observed issue, where predicted values lack spa- tial variability and fail to preserve local extremes,limiting their usefulness in tasks such as extreme event detection or fine-grained forecast correction. To better understand the origin of these challenges. we conduct a detailed empiri- cal analysis in Appendix [D] To address these issues, we propose DiffLiG (Diffusion- -enhanced Liquid Graph with Attention Propagation), graph neural network framework specifically designed to model station heterogeneity and alleviate oversmoothing in station-level interpolation.
GeoLiquidNNtt
Dynamic Edge Modulator
X
Heterogeneity- & Oversmoothing--miiiee Output
Figure 1: Two core challenges in station-level interpolation. Hetero- geneity across stations is addressed by the GeoLiquidNet, which models site-specific temporal dynamics, and the Dynamic Edge Modulator, which learns spatially adaptive graph struc- tures. Oversmoothing is mitigated by the Probabilistic Diffusion Selector, which generates and selects diverse cor- rection candidates to preserve structural variability.
To overcome these challenges, DiffLiG introduces uni- fied, uncertainty-awaa aaware interpolation framework composed of three key components: GeoLiquidNet, which com- bines temporal attention over gridded inputs with a site- aware liquid neural network to capture station-specific temporal dynamics; a Dynamic Edge Modulator, which adaptively configures spatial graphs by jointly learning edge existence, directional attention, and distance-aware influence; and Probabilistic Diffusion Selector, which generates multiple candidate corrections through noise- perturbed graph propagation and selects aggregates them via a context-aware scoring mechanism, thereby en- hancing both structural variability and predictive reliability
29
3
In summary, our contributions are as follows:
We propose DifTLiG, a novel graph neural network framework designed to bridge the gap between gridded forecasts and station-level observations, targeting station heterogeneity and oversmoothing.

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We design three specialized modules to address station heterogeneity and prediction oversmoothing: GeoLiquidNet for adaptive temporal encoding based on attention and site-aware OU dynamics; Dynamic Edge Modulator for configuring spatially adaptive graphs through directional attention and learnable edge selection; and Probabilistic Diffusion Selector for generating diverse correction candidates and selecting reliable outputs under uncertainty.
Extensive experiments across multiple forecasting sources, variables, and geographic regions demonstrate that DiffLiG consistently outperforms traditional interpolation and GNN baselines in both aceuracy and robustness at the station level.
2 Related Work
Grid-based forecasting. Grid-structured modeling is the backbone of both numerical and AI- driven weather prediction systems. Operational centers such as ECMWF deploy deterministic high-resolution forecasts (IFS-HRES) and ensemble-
systems (IFS-ENS) by solving physical equations over structured grids. Recent Al models have inherited this gridded formulation: FourCast- Net (30) employs Fourier neural operators for global fields; GraphCast [22 uses spherical GNNs for medium-range forecasts; Pangu- Weether [5] introduces 3D convolutional architectures for volumetric prediction. Other large-scale efforts include FuXi [IO] and Fengwu [ which are trained on global reanalysis datasets and have demonstrated competitive performance in medium-range forecasting tasks. Foundation models such as AtmoRep 24 Aurora O and ClimaX (28] further decouple forecasting from supervision by pretraining over vartilva grid datasets. However, these models produce outputs aligned to fixed grid structures and often struggle adapt to sparse, topography- sensitive station layouts. This mismatch limits their direct applicability in real-world settings where localized accuracy critical. Our work bypasses the grid altogether, targeting station-level prediction with geometry-awan spatial adaptation.
From grids to stations. Traditional spatial interpolation methods, such inverse distance weighting (IDW), bilinear interpolation, kriging, and spline interpolation, efficiently generate point forecasts from gridded fields, require no training data, and offer clear interpretability with low computational cost. More recently, learning-basse based approaches have sought to improve accuracy and capture complex dependencies: Bentsen al. < apply a GNN to station time series for wind prediction, MetNet-3 2 fuses satellite and station inputs via a transformer- style U-Net (albeit on regridded km mesh), Zhao al. [4] further highlight the smoothing bias in ERA5 during extreme tropical cyclone rainfall events and propose deep learning- based correction framework, and MGNN 39] combines ERA5 and MADIS in multimodal graph for deterministic forecasts. However, all these methods either depend on grid-aligned inputs outputs, fail to adapt to station-specifie heterogeneity, and lack mechanisms for uncertainty quantification or structural diversity. In contrast, our framework operates natively off-grid, leveraging geometry-awe graph with adaptive edge modulation and geo-liquid temporal dynamics, together with diffusion- -based ensemble generation to preserve fine-scale variability and capture predictive uncertainty.
3 Method
3.1 Problem Formulation
We address the problem of spatial interpolation from structured gridded meteorological forecasts to irregularly distributed observation stations. Our goal is to develop a lightweight and scalable graph neural network (GNN) framework that ensures physical consistency, strong spatial generalization. and the ability to model predictive uncertainty.
Formally, the model takes as input triplet (.).
X (x(9) E CxH -=- 0, ..... +.2) denotes the gridded forecast data over T = = +T+ - time steps. Each (*) is a multichannel spatial field (c.g., precipitation, wind) with C' variables over regular grid of size H x W (c.g., 721 x 1440 for 0.25 global resolution).
P = {p3 E j= M) represents the spatial coordinates (latitude and longitude) of M observation stations.
3

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= 1 j = ... M; t contains the historical observations at
each station over T + time steps (from -7i 0).
The objective is to learn mapping function:
F (x, P. 7) - y. where y [ū E R 1,..., M}
that predicts the target variable each station for the future time step t +12. The model is trained to minimize the prediction error with respect to the ground truth:
Úg = Vj (+Tz)
11......, M.M
3.2 DiffLiG Overview
Nl S(b,3)
Probabilittic
Aftentian
Maitc Malt--hllria Se Stare Aptragaterr
Station Locationa(")
GCN
Histerical Banagh)
GeoLiquidNet
Matulaon
Dynamic Ediee
MLP
Vemporal Atication MLLP
Bls
00 00
I'mbedding
Station
Weights
Precipitadion Value
Station Features
Gribded Fomeaat())
Dynamic Edge
Probabilistic
Modulator
Forecast Diffuser
Edge Gating
Gvographic Decny
Attention Net
Noise
Overview of DiffLiG. The model takes gridded metcorological forecasts X, station coordi- nates P, and historical station observations H as input. The EeoLiquidNe module encodes temporal features for each station based on H and P. The Forecast Encoder assigns temporal attention weights across different time steps of X, followed by embedding MLP to extract forecast representations. These representations are integrated by the Dynamic Edge Modulator, which constructs adaptive graph edges through edge gating, attention net, and geographic decay. The Probabilistic Forecast Diffuser generates multiple perturbed predictions to capture epistemic uncertainty. A Multi-Criterie Forecast Selector evaluates and selects the most reliable prediction among them. Finally, the selected prediction is refined by the Station Bias Module, which incorporates site-specific adjustments (+Ta)
Figure 2:
through output MLP, producing the final forecast y (oy ) where ún se y
3.3 GeoLiquidNet
To handle the non-stationary nature of precipitation sequences, where relevant timescales vary across weather systems [6)], regions, and station density, we propose GeoLiquidOU, module that learns spptiaaly-conddttioneed time constants to adaptively control hidden state transitions. Inspired by Liquid Time-constant Networks (LTC) [15] which model time constants input-dependent variables to capture nonlinear dynamics, GeoLiquidOU inherits this property to adaptively modulate temporal sensitivity based on each site's historical inputs. Neuroscientific insights further support this design: the diversity of timescales in the brain reflects the dynamic nature of encoded information [1]. analogous to varying temporal patterns in meteorological data. GeoLiquidOU thus integrates dynamic temporal encoding with location-awas adaptability, enhancing both interpretability and flexibility in modeling site-specific temporal patterns. Further analytical discussions on the OU-based dynamics deferred to the Appendix

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Formally, for station with imput sequence h =
...., and location R², the module first embeds each timestep via 2(t) = pruj (f)()), where Opnj( ()) is learnable projection
MLP that maps the raw observation ((() into latent feature space.
learnable location encoder Tpp) then generates the step controller as:
T Sigmoid 1 (W)) + 0.5,
where W1 and W2 are linear transformation matrices. The output 7 (p) serves as a positive location- dependent scaling factor that defines site-specific time interval A/ r(p) Ahhas-
The latent state ž, E IR evolves through a parameterized Ornstein-Uhlenbeck (OU) process:
Az. At [7. (0(z)))) (0(2)))))))))
21) t
(I)
where (-) is a learnable MLP that predicts the target equilibrium state, o E Rd are trainable dynamic coefficients controlling the deterministic and stochastic components, and e ~ () denotes Gaussian noise.
The process is initialized with Zu Omm (h()). and iteratively updated over T steps. The final state Zr thus encodes the historical evolution of each site, capturing temporal dynamics modulated by its geographic location. This continuous-time formulation allows GeoLiquidNet flexibly adapt to diverse temporal rhythms across stations, enhancing generalization and interpretability in sequence modeling.
3.4 Spatial Message Passing Network
To model the spatial heterogeneity and directional propagation of meteorological variables, we introduce Spatial Message Passing Network (SMPN). It adaptively learns graph structure and edge importance through four components: edge selection, distance-aware decay, directional attention, and residual node update.
Edge Selection Mechanism. We employ a neural discriminator (=()) to predict the existence proba- bility of an edge based node coordinates and their Haversine distance.g,; g [p: Py. d.yl), where G(-) denotes a learnable gating function that outputs an edge existence probability. During training, binary gate Oiy E {0, 1} is sampled from Pi enable data-driven, spatially adaptive edge pruning 12
Distance-aware Decay.
To reflect spatial attenuation, each edge is weighted by decay factor Ki = (xxp(-Bd,)) where dy is the Haversine distance. The decay rate 3 is learned via bounded sigmoid function, allowing smooth, interpretable modulation of long-range influence.
To capture directional dependencies, design an interaction- attention mechanism (37 40, 8.. For each edge, the input vector e;₃ = [h, ||(,: P,)] is processed by
Multi-head Attention.
M attention heads. Normalized scores a (m) determine the contribution of neighbor enabling asymmetric and context-sensitive aggregation 42
Residual Update. The final propagation weight is wij - Dej Kaj Ouj. Messages mij are generated via @msg and aggregated with normalization. Each node is updated via a residual MLP [ [6]
h h, Gupd (l,,, m.))
This enables stable, fine-grained updates across two stages: first from gridded fields stations, then among stations for local refinement.
5

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Algorithm 1: Spatial Message Passing Network
Input: Node features h, positions pi, number of spatial neighbors k, edge gates Pij. decay factors Kijs attentions Okij
1. For each node i, identify the l nearest neighbors (x based on Haversine distance
2. Initialize m. +- 0, count +
3. For each j E (i):
If 9tj = compute propagation weight way = 9ij Kij aij Compute message mij = cmes (lh,, hy, P; - P;l))
Accumulate: m; += Wij maji increment count
4. Normalize: m,, <- /couu
5. Update node state: h h2 + Pupdl [ba, m.l)
3.5 Probabilistic Forecast Diffuser
To capture the inherent uncertainty of meteorological predictions and generate diverse forecast candi- dates, propose Probabilistic Forecast Diffuser. This module injects stochastic perturbations into node features and processes them through a graph-based denoising network, yielding set of sample-level forecasts for subsequent evaluation and selection. A concise mathematical analysis explaining why the diffusion process enables recovery and amplification of extreme precipitation values is provided in Appendix F
Sample Generation via Difflusion. Given node features and position encodings @(p), we generate N samples by adding Gaussian noise ~ (O, o²I) and passing the perturbed inputs through graph U-Net denoiser fuNe
18 35
Each sample is constructed via residual correction:
8 = A x (1 A) - (a + - fuNa(1 + +& o(p)))
(2)
Here, E [0, 1] a blending coefficient that controls the contribution of original versus corrected features. The result is ensemble of spatially coherent yet diverse forecasts (81,...,8N)
Uncertainty Estimation. We quantify prediction uncertainty for each node by computing the sample variance:
1 N
N
1
Uj
= N
Σ
8,)
Si.j
N
1=1
1=1
This serves as a natural metric for identifying regions of disagreement across the ensemble and guiding adaptive selection.
3.6 Multi-Criteria Score Selector
To extract the most plausible forecast from the generated ensemble, or combine multiple outputs into robust final prediction, we introduce the each candidate using three scoring perspectives: semantic consensus, spatial coherence, and physical plausibility [21 25
Score Selector. This module evaluates
Semantic Consensus. A self-attention mechanism is applied across all N samples to estimate their mutual agreement. Each sample assigned an attention-based consensus score, measuring its alignment with the ensemble majority.
2. Spatial Coherence. To enforce geophysical smoothness, we pass each sample through lightweight graph convolution network that estimates the degree of spatial consistency across neigh- boring stations. This captures the natural continuity of meteorological variables in space.
Extreme Value Detection. We implement lightweight MLPe- scoring head that focuses on detecting physically implausible or statistically extreme predictions. It penalizes samples that contain abnormal spikes, unrealistic magnitudes, or values significantly deviating from historical ranges, thereby reducing their influence during ensemble selection.

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Final Selection. The above three scores are aggregated using two-layer MLP to compute a unified quality score q; for each sample. We apply softmax over all scores to produce sample weights. The final output is selected via either hard picking (the best-scoring g:) or weighted fusion of the ensemble.
4 Experimental
4.1 Experimental Setup
To comprchensively evaluate the generalization and robustness of the proposed method under various meteorological models and complex geographical conditions, we design a station-level precipitation interpolation task based on multiple forecast products and ground truth observations.
Data Sources. We utilize 24-hour accumulated precipitation forecasts from five representative meteorological models, including both traditional numerical systems and emerging Al-based models. Specifically, the selected inputs cover: ECMWF's high-resolution deterministic forecast (FF-H, ensemble forecast system (IFS-ENS), and reanalysis data (ERA5), well as models such FourCast- Net, and GraphCast, As ground-truth reference, we employ in-situ measurements from 2288 surface meteorological stations provided by the China Meteorological Administration. These stations distributed across diverse terrains and climatic regions, introducing strong spatial heterogeneity. All observational data are resampled to daily resolution and temporally aligned with forecast inputs. The resulting dataset forms a 24-hour accumulated precipitation interpolation benchmark for evaluating spatial reasoning and correction capabilities under complex landscapes.
Task Definition. consists of coarse-resolution gridded imputs and historical observations at stations, and the model is required to predict the target variable (e.g., 24-hour precipitation) at all station locations.
The task is formulated as a trid-to-station spatial mapping problem. Each sample
Extended Experiments. To examine the model's variable adaptability and temporal generalization, we also conduct extended experiments on 2-meter temperature prediction using Pangu-Was Weather temperature products. Detailed results of this auxiliary experiment included in the Appendix G We report additional results in the Appendix H including a sensitivity analysis under varying forecast horizons and structural parameter settings.
Category
Methods
Model
P (#)
S (MB)
Traditional Interpolation
Nearest Neighbor
DinTLiG (ours)
49.361
0.19
Linear Interpolation
LiGAP
39.397
0.15
Inverse Distance Weighting
GAP
33,650
0.13
Ordinary Kriging
Graph-bias
29,506
0.11
Graph Neural
GCN 19
Graph
24,930
0.10
Networks
38
GCN
29,031
0.11
GraphSAGF 14
GAT
29,287
0.11
KCN 3
GraphSAGE
33,127
0.13
KCN
30,520
Physics-Inspired
MoshGraphNet
3]
Architectures
MGNN
VIT
51,538
0.20
Diffusion
54,014
0.21
Spatial & Generative
VIT III
MeshGraphNNe
41,863
0.16
Diffusion Networks
MGNN
40,741
0.16
Table 1: Classification of Baseline Models
Table 2: Model Complexity Comparison
Baseline Models. To assess model performance, we compare against a broad range of interpolation and learning-basse baselines, as shown in Table This diverse benchmark suite allows us to evaluate accuracy, structural consistency, and generalization ability across multiple paradigms. A more detailed description of the baseline methods, including their modeling assumptions and limitations, is provided in the Appendix K
Model Complexity. To further examine architectural efficiency, we report the parameter counts and storage sizes of all compared models in Table 2 Although DiffLiG is not the smallest in scale, it introduces only a moderate increase parameters while achieving significant gains accuracy and generalization, reflecting efficient use of model capacity. Its ablation variants reveal the role of ensemble component from DiffLiG, GAP
individual modules: LiGAP removes the further discards the Temporal Module to isolate temporal contributions, Graph-bias omits the Spatial
7

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Message Passing Network, and Graph represents the most reduced configuration without the Station Bias Module.
Evaluation Metrics. We employ five quantitative metrics to comprehensively assess model perfor- mance: Root Mean Squared Error (RMSE), Anomaly Correlation Coefficient (ACC), and Threat Scores (TS) at 2 mm, 20 mm, and 100 mm thresholds, RMSE measures the overall magnitude devia- tion between predictions and observations, ACC evaluates spatial correlation and pattern consistency across stations, while TS metrics quantify the model's capability to detect precipitation events of increasing intensity, reflecting its sensitivity light, moderate, and extreme rainfall. Together, these metrics provide a balanced evaluation of numerical accuracy, spatial coherence, and event-level detection skill.
Implementation and Training Details. To ensure fair comparison, all baseline models are trained under identical hardware environments, using the same loss function and optimization settings. Further experimental details, including training strategies, configuration,, and runtime statistics, are provided in Appendix [
4.2
Experimental Results
IFS-HRES
ERA5
IPS-ENS
FourCastNet
Method
RMSE
ACC
T5@2mm
RMSE
ACC
TS
RMSE
ACC
TS022m
ACC
7mmm
DiffLIG (ours)
5,0883
0.7395
0.4621
4.8021
0,7677
0.5121
4,7208
0.7721
0.5286
5,1014
0.7303
0.4937
Ncarest
6,0898
0.6511
0.1121
5.2145
0.7265
0.4693
1.9717
0.7468
0.1801
5.3871
0.6795
0.1865
Linear
5.9033
0.6667
0.4011
5.1435
0.7313
0.4808
4.9395
0.7504
0.4757
5.3721
0.6948
0.4858
OK
5.7423
0.6577
0.3931
5,1011
0.7387
0.4891
4.8922
0,7581
0.4451
5,3088
0.7052
0.1811
IDW
5.8274
0.6742
0.39915
5.1058
0.7368
0.4558
4.9299
0.7515
0.4716
5.3683
0.6531
0.4837
GCN
5.4832
0.6927
0.41339
5.0G80
0.7417
0.4385
4.8731
0.7610
0.5052
5,2313
0.7016
0,4970
GAT
5.5444
0.6867
0.32812
5.0400
0.7440
0.4388
4.9137
0.7572
01.4977
5.255
0.6964
0.4521
GraphSAGE
5.6101
0.6195
0.3754
5.1154
1.9343
0.7423
5,2511
0.7001
0,1936
KCN
5.3931
0.6962
0.1158
5.0897
0.7468
09.4998
1.5897
0.7572
0.4925
5,2264
0.7122
0.4929
VIT
3.7848
0.6333
0.3626
5.1232
0.7121
0.4428
5.2365
0.7034
0.4832
5.3912
0.7080
0.4812
Dillusion
0.5963
0,3342
5,1001
0.7284
0.4537
5.1742
0.7188
0.4952
5.3781
0.6932
0.4921
MeshGraphNet
5.5322
0.67677
0.43744
5.1728
0.7283
0.4903
5.0132
0.7332
0.5144
5.2619
0.6931
0,4987
MGNN
5.4428
0.6858
0,3640
4,9904
0.7501
0,5000
4,8542
0,7652
0.4357
5,1878
0,7164
0.4123
Table 3: Performance Comparison of Interpolation Methods across Multi-Source Forecast
Inputs. Top-3 values in each metric are highlighted with increasing color intensity. Bold values indicate the best performance, while underlined values represent the second-best results for each metric.
To evaluate the model's capability in extracting reliable station-level information from coarse- resolution gridded forecasts, we conduct comprehensive experiments across four representative forecast sources: IFS-HRES, ERAS, IFS-ENS, and FourCastNet. Results are summarized in Table 3
DifTLiG demonstrates consistent superiority across all three metrics. In terms of TS @ 2mm, which measures the model's s ability to identify precipitation events, DiffLiG achieves the highest scores on IFS- (0.4621), ERA5 (0.5121), and IFS-ENS (0.5286). On FourCastNet, it reaches 0.4937, closely following the best-performing baseline (0.4987), underscoring its strong event detection capability. For RMSE, DiffLiG achieves the lowest error across all four sources, indicating more accurate magnitude reconstruction. In terms of ACC, DiffLiG again attains the highest accuracy on each dataset, reflecting its consistent predictive correctness and classification reliability.
Furthermore, the consistent improvements of DiffLiG over all baselines across multiple evaluation metrics suggest its strong capacity to mitigate station-level heterogeneity. Unlike traditional methods that often suffer from degraded performance in regions with complex topography or sparse obser- vations, DiffLiG achieves lower RMSE (more accurate magnitude), higher ACC (stronger spatial correlation with observations), and higher T 2mm (better event detection skill). These results indi- cate that the model effectively adapts to diverse spatial characteristics and observational conditions, demonstrating its practical utility for heterogeneous station-level interpolation in real-world settings.
In addition, DiffLiG exhibits strong cross-source generalization. It maintains either top or second- best performance across all forecast sources, regardless of the underlying modeling paradigm: be it physically based numerical weather prediction (e.g., IFS-HRES, IFS-ENS) or Al-driven models (e.g., FourCastNet). This cross-domain consistency validates the robustness of its architectral design under varying input distributions and error characteristics.
8

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4.3 Evaluation of Oversmoothing Mitigation on IFS-HRES
To evaluate capability of different models in mitigat- ing spatial oversmoothing, we compute TS at 20mm and 100mm thresholds on the IFS-HRES datasct. As shown in Table DiftLiG achicves the highest TS values across both thresholds, demonstrating its effectiveness in alle- viating the oversmoothing problem. LiGAP, which re- moves the diffusion module from DiffLiG while keeping all other components identical, shows a noticeable degra- dation both TS @ 20mm and TS @ 100mm, highlighting the effectiveness the probabilistic diffusion mechanism mitigating the oversmoothing effect inherent graph propagation.
Model
T 20mm
TS@ 100mm
DifILiG (ours)
0.3122
0.0202
LiGAP
0.2819
0.0147
Linear
0.2764
0.0156
GAT
0,2986
0,0162
GCN
0,2836
0.0131
GraphSAGE
0.2648
0.0157
KCN
0.2862
0.0136
MeshGraphNet
0.2925
0.0144
MGNN
0.2611
0.0153
Table 4: Oversmoothing Mitigation IFS- HRES.
4.4
Visual Evidence of Oversmoothing Mitigation
Oberurvation
Interpolation
LeGAP
biffl (
Area
Figure 3: Visual comparison of spatial interpolation results. Predicted daily accumulated precipi- tation for July 2022.
To assess oversmoothing mitigation, we visualize predicted station-level precipitation on July 5, 2022-a a day with high spatial variability (Figure 3). Compared to baseline methods, DiffLiG produces sharper and more spatially coherent outputs, capturing localized extremes more effectively. LiGAP, which removes the diffusion module from DiffLiG, shows improved spatial structure but remains conservative due to its deterministic nature. The full DiffLLi model leverages ensemble diversity and selective refinement to better recover fine-scale variability aligned with observations. A more detailed visualization of the results is provided in the Appendix N for further reference.
4.5 Cross-Dataset Evaluation on GraphCast for Robustness Validation
GraphCast
DifLIG
Method
RMSE
ACC
TS@:2mm
DifILIG (ours)
5,5172
0.7345
0,5218
Nearest
5.7036
U.7133
0.3520
Linear
5.6829
0.7153
0.3515
OK
5.5923
0.7132
0.3612
IDW
5.6772
0.7138
0.3506
GCN
5,5517
0.7302
0.4985
GAT
6.5773
0.7271
0.5245
GrapbSAGE
5.8757
0.6811
KCN
5.5980
0.7256
0.5181
VIT
5.8321
0.6844
0.1212
Diffusion
5.7342
0.6923
0.4354
MeshGraphNet
6.2677
0.6246
0.5171
MGNN
5.6537
0.7288
0.3526
Table 5: Performance of different methods on GraphCast.
Figure 4: Spatial distribution (left) and scatter-
plot (right) for DiffLiG vs. GCN.

---

To evaluate the robustness and generalization capability of our heterogeneity- aware forecasting model, we conduct a cross cross-dttaset inference experiment. Specifically, the model trained entirely on the IFS-HRES dataset and directly applied to the GraphCast dataset without any fine-tuning or domain adaptation. The performance comparison is reported in Table 5 and the spatial error distribution alongside station-wise scatter-plot of DiffLiG versus GCN is visualized in Figure
4.6 Ablation Studies
Structural modeling. Starting from the base model with static edges and uniform aggregation, perfor- mance limited across all metrics. Adding the
Performance Improvement with Model Components
Allation Study: RMsse
Dynamic Edge Modulator leads to significant im- provement TS@ @ 2mm, highlighting the benefit of learning spatially adaptive connections. The Station Bias Correction further improves ACC by captur- ing site-specific deviations, especially where grid forecasts systematically diverge from observations. These modules enhance the model's 's ability to reflect spatial heterogeneity, but temporal dynamics remain unmodeled.
5-4233
125h
14775
2010
100
Study, Acairacy (ACC)
168%
1N
0.093
Temporal encoding. Introducing the Temporal Mod- ule leads to substantial improvement in TS @ 2mm (+16.3%) indicating enhanced sensitivity to local- ized rainfall events. By jointly encoding gridded sequences and station histories, the model better cap- tures evolving precipitation patterns, leading to con- current improvements both ACC and RMSE.
Ablation Study: Threat Score (T502mm)
HUS
0,993
Uncertainty modeling and selection. The Proba- bilistic Forecast Diffuser reduces RMSE by modeling ensemble uncertainty, though basic aggregation meth- ods show limited effect on TS and ACC. The final addition of the Multi- CritirCiieeia Score Selector brings all metrics to their best levels by prioritizing forecasts with spatial consistency and physical plausibility. This confirms the importance of informed selection in ensemble-based interpolation.
Figure 5: Ablation trend. Module-wise improvements across RMSE, ACC, and @2@2.m.
5 Conclusion
We presented DiffLiG, a diffusion eenfancd liquid graph framework for high- high-resolution spatial correction from gridded forecasts to station-level observations. To address the challenge of station heterogeneity, DiffLiG integrates a Geo-Liquid Net that captures site-specific temporal dynamics, and Dynamic Edge Modulator that independently learns spatially adaptive graph structures tailored to each target station. To mitigate oversmoothing and model predictive uncertainty, a unified Probabilistic Diffusion Selector generates diverse correction candidates via noise-perturhed sampling and refines them through multi-criteria ensemble selection.
Extensive experiments across multiple meteorological datasets and forecast products demonstrate that DiffLiG delivers consistent improvements in both accuracy and event sensitivity over classical in- terpolation methods and strong graph-based based baselines. Its effectiveness extends beyond precipitation, achieving strong performance on other key variables such as temperature, highlighting its flexibility and generalizability.
DiffLiiG offers a unified and efficient framework for correcting coarse meteorological forecasts under real- world spatial complexity, with strong potential for application in post-processing, data assimilation, and multi-source fusion tasks. Looking forward, DifILiG provides a foundation for exploring graph-based spatial correction under more complex scenarios, such cross-regional generalization, extreme weather adaptation, and integration with multi-modal inputs (e.g., satellite imagery or radar). Future work may also incorporate causal or physical priors to further enhance the interpretability and resilience of correction under evolving climate dynamics.
10

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Acknowledgements
This work was supported in part by the National Key Researchand Development Plan of China under Grant F033YB33002000; in part by the National Natural Science Foundation of China under Grant T2125006 and Grant 42401415; in part by Shenzhen Science and Technology Program under Grant XCXF72002400030335500 and Grant KJZDD02309231 15 106012; 2; in part by the Open Research Fund of Pengcheng Laboratory under No.2222KKFIB0400 and part by Jiangsu Innovation Capacity
Building Program under Project B.2022028..
References
[1] Martin Andrae, Tomas Landelius, Joel Oskarsson, and Fredrik Lindsten. Continuous ensemble weather forecasting with diffusion models, 2025.
[2] Marcin Andrychowicz, Lasse Espeholt, Di Li, Samier Merchant, Alexander Merose, Fred Zyda, Shreya Agrawal, and Nal Kalchbrenner. Deep learning for day forecasts from sparse observations, 2023.
[3] Gabriel Appleby, Linfeng Liu, and Li-Ping Liu. Kriging convolutional networks. Proceedings of the AAAI Conference on Artificial Intelligence, 34(04):3187-3194 April 2020.
[4] Guido Ascenso, Andrea Ficchi, Matteo Giuliani, Enrico Scoccimarro, and Andrea Castelletti. Downscaling, bias correction, and spatial adjustment of extreme tropical cyclone rainfall in era5 using deep learning. Weather and Climate Extremes, 100744 2024.
[5] Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian. Pangu-weather: A 3d high-resolution model for fast and accurate global weather forecast, 2022.
[6] Cristian Bodnar, Wessel P. Bruinsma, Ana Lucic, Megan Stanley, Anna Vaughan, Johannes Brandstetter, Patrick Garvan, Maik Riechert, Jonathan A. Weyn, Haiyu Dong, Jayesh K. Gupta, Kit Thambiratnam, Alexander Archibald, Chun-Chieh Wu, Elizabeth Heider, Max Welling, Richard E. Turner, and Paris Perdikaris. A foundation model for the earth system, 2024.
[7] Xavier Bresson and Thomas Laurent. Residual gated graph convnets, 2018.
[8] Shaked Brody, Uri Alon, and Eran Yahav, How attentive are graph attention networks?, 2022.
[9] Kang Chen, Tao Han, Junchao Gong, Lei Bai, Fenghua Ling, Jing-Jia Luo, Xi Chen, Leiming Ma, Tianning Zhang, Rui Su, Yuanzheng Ci, Bin Li, Xiaokang Yang. and Wanli Ouyang. Fengwu: Pushing the skillful global medium-range weather forecast beyond 10 days lead, 2023.
[10] Lei Chen, Xiaohui Zhong, Feng Zhang, Yuan Cheng, Yinghui Xu, Yuan Qi, and Hao Li. Fuxi: cascade machine learning forecasting system for 15-day global weather forecast, 2023.
[11] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at scale, 2021.
[12] Luca Franceschi, Mathias Niepert, Massimiliano Pontil, and Xiao He. Learning discrete structures for graph neural networks, 2020.
[13] Peter Grönquist, Chengyuan Yao, Tal Ben-Nun, Nikoli Dryden, Peter Dueben, Shigang Li, and Torsten Hoefler. Deep learning for post-processing ensemble weather forecasts. Philosophi- cal Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379(2194):20200992, February 2021.
[14] William L. Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs, 2018.
[15] Ramin Hasani, Mathias Lechner, Alexander Amini, Daniela Rus, and Radu Grosu. Liquid time-constant networks, 2020.

---

[16] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition, 2015.
[17] Hans Hersbach, Bill Bell, Paul Berrisford, Shoji Hirahara, András Horányi, Joaquín Muñoz- Sabater, Julien Nicolas, Carole Peubey, Raluca Radu, Dinand Schepers, Adrian Simmons, Cornel Soci, Saleh Abdalla, Xavier Abellan, Gianpaolo Balsamo, Peter Bechtold, Gionata Biavati, Jean Bidlot, Massimo Bonavita, Giovanna De Chiara, Per Dahlgren, Dick Dee, Michail Diamantakis, Rossana Dragani, Johannes Flemming, Richard Forbes, Manuel Fuentes, Alan Geer, Leo Haimberger, Sean Healy, Robin J. Hogan, Elías H6lm, Marta Janisková, Sarah Keeley, Patrick Laloyaux, Philippe Lopez, Cristina Lupu, Gabor Radnoti, Patricia de Rosnay, Iryna Rozum, Freja Vamborg, Sebastien Villaume, and Jean-Noël Thépaut. The era5 global reanalysis. Quarterly Journal of the Royal Meteorological Society,
2020.
[18] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models, 2020.
[19] Bo Jiang, Ziyan Zhang, Doudou Lin, and Jin Tang. Graph learning-convolutionall networks, 2018.
[20] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017.
[21] Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles, 2017.
[22] Remi Lam, Alvaro Sanchez-Gonzalez, Matthew Willson, Peter Wirnsberger, Meire Fortunato, Ferran Alet, Suman Ravuri, Timo Ewalds, Zach Eaton-Rosen, Weihua Hu, Alexander Merose, Stephan Hoyer, George Holland, Oriol Vinyals, Jacklynn Stott, Alexander Pritzel, Shakir Mohamed, and Peter Battaglia. Graphcast: Learning skillful medium-range global weather forecasting, 2023.
[23] Erik Larsson, Joel Oskarsson, Tomas Landelius, and Fredrik Lindsten. Diffusion-lam: Proba- bilistic limited area weather forecasting with diffusion. In ICLR 2025 Workshop Tackling Climate Change with Machine Learning, 2025.
[24] Christian Lessig, Ilaria Luise, Bing Gong, Michael Langguth, Scarlet Stadtler, and Martin Schultz. Atmorep: A stochastic model of atmosphere dynamics using large scale representation learning, 2023.
[25] Yaguang Li, Rose Yu, Cyrus Shahabi, and Yan Liu. Diffusion convolutional recurrent neural network: Data-driven traffic forecasting, 2018
[26] Siqi Liu and Andreas Lehrmann. Dynaconf: Dynamic forecasting of non-stationary time series, 2024.
[27] S. Karthik Mukkavilli, Daniel Salles Civitarese, Johannes Schmude, Johannes Jakubik, Anne Jones, Nam Nguyen, Christopher Phillips, Sujit Roy, Shraddha Singh, Campbell Watson, Raghu Ganti, Hendrik Hamann, Udaysankar Nair, Rahul Ramachandran, and Kommy Weldemariam. Ai foundation models for weather and climate: Applications, design, and implementation, 2023.
[28] Tung Nguyen, Johannes Brandstetter, Ashish Kapoor, Jayesh K. Gupta, and Aditya Grover. Climax: A foundation model for weather and climate, 2023.
[29] Joel Oskarsson, Tomas Landelius, Mare Peter Deisenroth, and Fredrik Lindsten. Probabilistic weather forecasting with hierarchical graph neural networks. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomezak, and C. Zhang, editors, Advances in Neural Information Processing Systems, volume 37, pages 41577-41644 Curran Associates, Inc., 2024.
[30] Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadeneshelii Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar. Fourcastnet: A global data- driven high-resolution weather model using adaptive fourier neural operators, 2022.
[31] Tobias Pfaff, Meire Fortunato, Alvaro Sanchez-Gonzalee and Peter W. Battaglia, Learning besh-based simulation with graph networks, 2021.
12

---

[32] Thomas Rackow, Nikolay Koldunov, Christian Lessig, Irina Sandu, Mihai Alexe, Matthew Chantry, Mariana Clare, Jesper Dramsch, Florian Pappenberger, Xabier Pedruzo-Bagazaee Bagauzzoiaaa Steffen Tietsche, and Thomas Jung. Robustness of ai-based weather forecasts in changing climate, 2024.
[33] Vivek Ramavajjala and Peetak P. Mitra. Verification against in-situ observations for data-driven weather prediction, 2023.
[34] Markus Reichstein, Gustau Camps-Valke Valls, Bjorn Stevens, Martin Jung, Joachim Denzler, and Nuno Carvalhais. Deep learning and process understanding for data-driven earth system science. Nature, 5667773) 2019.
[35] Olaf Ronneberger, Philipp biomedical image segmentation, 2015.
Fischer, and Thomas Brox. U-net: Convolutional networks for
[36] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations, 2021.
[37] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need, 2023.
[38] Petar Veličkovié, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks, 2018.
[39] Qidong Yang, Jonathan Giezendanner, Daniel Salles Civitarese, Johannes Jakubik, Eric Schmitt, Anirban Chandra, Jeremy Vila, Detlef Hohl, Chris Hill, Campbell Watson, and Sherrie Wang. Local off- off-grid weather forecasting with multi-modal earth observation data, 2025.
[40] Seongjun Yun, Minbyul Jeong, Raehyun Kim, Jaewoo Kang, and Hyumwoo J. Kim. Graph transformer networks, 2020.
[41] Roxana Zeraati, Anna Levina, Jakob H. Macke, and Richard Gao. Neural timescales from computational perspective, 2024.
[42] Kai Zhang, Yaokang Zhu, Jun Wang, and Jie Zhang. Adaptive structural fingerprints for graph attention networks. In International Conference on Learning Representations, 2020.
[43] Jiachen Zhu, Xinlei Chen, Kaiming He, Yann LeCun, and Zhuang Liu. Transformers without normalization, 2025.
[44] Lars Ødegaard Bentsen, Narada Dilp Warakagoda, Roy Stenbro, and Paal Engelstad. Spatio- temporal wind speed forecasting using graph networks and novel transformer architectures. Applied Energy, 33561, 2023.
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Appendices Contents
Broader Impact
22
Limitations
22
Empirical Analysis of ERA5 Precipitation Interpolation Errors
23
D
Origins of Station Heterogeneity and Oversmoothing Biases
25
Theoretical Analysis of OU-Based Temporal Encoding
25
E.
Continuous-Time OU Dynamics
25
E.2
Discrete-Ting Approximation via Buler Method
26
................................
E.3
Explicit Recursion and Temporal Weighting
................................
26
Prequency-Domall Interpretation: A Low-Pas Pass Filter
26
E.S
Interpretation for Geol.iquidOU
27
atenttSpacce Noise Injection and UNet Denoising with Residual Perturbation
27
Cross- Variable Forecasting: 'empperturre as Input
28
Sensitivity Forecast Horizon and Graph Connectivity
30
H.I
Sensitivity to Forecast Lead Time
30
H.2 Impact of Backtracking Window Size
31
H.3 Impact of ERA5-to- Edge Connectivity
Impact of Station-to-Statiin Connectivity
33
implementation and Training Details
33
Training Objective and Loss Function
33
1.2
Task Configuration
33
1.3 Training Configuration
33
Staged Training Strategy
33
J
Haversine Distance Computation
33
K Baseline Method Descriptions
34
Evaluation Metrics
36
M License and Attribution Details
36
Visual comparison of spatial interpolation results
37
21

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A
Broader Impact
This study proposes lightweight and modular graph-based framework for station-level precipitation correction, designed to enhance the adaptability and spatial resolution of existing weather forecasting systems. Without requiring the retraining of large-scale models, our approach enables flexible post- processing of numerical or Al-generated forecasts, producing locally refined predictions at arbitrary coordinates using only location metadata and limited historical observations. This capability supports fine-grained, adaptive forecasting across diverse geographies, including remote, mountainous, or inder-instrumented regions, and has the potential to broaden the accessibility of reliable weather services worldwide.
Our method improves the spatial precision and reliability of extreme precipitation forecasts by explicitly modeling site-specific spatial structure, incorporating dynamic edge modulation, and introducing a diffusion-based ensemble generator with a multi-criteria output selector. This allows more accurate localization high-risk areas and robust identification of extreme-value signals, thus offering decision-level support for early warning systems related to flooding, torrential rainfall, or hydrological disasters. Unlike coarse-resolution grid-based outputs, the proposed framework allows targeted forecast refinement at critical sites, contributing to improved disaster preparedness and risk mitigation.
In terms of sustainability, the proposed architecture emphasizes post-hoc spatial adaptation rather than end-to-end retraining, reducing the demand for high-performance computing and enabling deployment in low-resource environments. As climate modeling and forecasting scale to global applications, energy-efficient and scalable forecasting solutions are urgently needed. Our model contributes to this trend by providing an extensible correction module that balances accuracy with computational economy.
Beyond meteorology, the modular design and uncertainty- structure of our model may benefit other spatiotemporal inference tasks with sparse measurements, such air quality estimation, ecological monitoring, or environmental hazard detection. The integration of spatial message passing, site heterogeneity handling, and mnsemble-based confidence evaluation offers transferable paradigm for building robust, site-adaptive models across scientific domains.
While we have not identified direct negative societal impacts associated with the proposed method, potential concerns may arise depending on downstream applications. For instance, if the model is deployed without adequate validation in regions with limited ground truth data, it may introduce misleading corrections or overconfident forecasts, especially in high-stakes scenarios. Moreover, localized correction systems, if misused or misinterpreted by non-experts, could inadvertently lead to false alarms or reduced trust in public forecasting services. These risks highlight the importance of responsible deployment and continual calibration against local observations when adapting the model to new domains.
B Limitations
While the proposed station-level correction framework demonstrates strong adaptability and predictive performance, several limitations remain that warrant further attention and refinement.
Our current implementation primarily utilizes precipitation and temperature variables from ERA5 as model inputs. Other meteorological factors that are physically relevant, such as wind speed, humidity, surface pressure, or convective potential, have not yet been incorporated. The exclusion of these features may limit the model's ability to capture complex atmospheric processes, particularly in scenarios driven by multi-varian variable interactions. Integrating additional variables, such as wind field structures or vertical profiles, into both the graph construction and node representation may enhance the model's responsiveness to high-impact weather systems.
Furthermore, although the model is generalizable to arbitrary geographic locations, our experiments have so far focused mainly on precipitation and temperature. Key atmospheric variables such as wind vectors or specific humidity have not yet been tested, and the current results not fully assess the framework's 's applicability in broader multi-modal forecasting scenarios. Different atmospheric variables exhibit distinet spatial dependencies and temporal dynamics, and whether the current modeling paradigm generalizes across such diversity remains open question.
22

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The model also assumes the availability of some historical station-level observations. Although we avoid retraining large-scale backbones, the quality and availability local data can still affect the performance of post-hoe corrections. In regions with highly sparse or irregular observations, such as new or mobile stations, the reliability of predictions may degrade. Enhancing model robustness through techniques such as transfer learning or structural priors could be promising future direction.
Lastly, while the diffusion-baseed ensemble generation and selection mechanisms provide a degree of uncertainty modeling, the current framework does not explicitly incorporate physical consistency constraints. Soft constraints based on hydrological or thermodynamic conservation laws (e.g. mois- ture or energy balance) could improve both the interpretability and reliability of outputs, especially under extreme or anomalous conditions.
C Empirical Analysis of ERAS Precipitation Interpolation Errors
To quantitatively motivate the need for high-resolution station-level correction, conduct system- atic assessment of precipitation biases present in the ERA5 reanalysis dataset after standard bilinear interpolation onto surface meteorological stations.
Spatial Distribution of ERAS Annual Precipitation Prediction Errors
ERAS Eroor damai day)
40
Latitude
900
20
Longitude
Figure 6: Spatial Distribution of ERA5 Annual Precipitation Prediction Errors. Each dot represents station. Color denotes the signed mean error (in mm/day) of bilinearly interpolated ERAS daily precipitation compared with ground observations, averaged over the entire year.
As shown in Figure 6 large spatial biases emerge across the region, with systematic overestimations in southwestern and southern China, and underestimations in southeastern coastal regions. These regional inconsistencies reflect the inability of grid-based interpolation adapt complex terrains and localized rainfall dynamics.
Figure further illustrates the severity of these discrepancies in relative terms. In regions such as western Sichuan and Yumnan, the relative errors exceed 75%, suggesting that the interpolated ERA5 data may drastically misrepresent local annual precipitation totals. This disproportionate bias is particularly problematic hydrologically sensitive regions or areas with sparse gauge density.
Figure X iillustrates the temporal variation of interpolation errors in ERA5 precipitation data across the year. The top panel shows that both RMSE and mean error rise significantly during the summer months, indicating reduced accuracy during the rainy season. The middle panel reveals that the maximum and absolute maximum daily errors also spike during this period, with values occasionally exceeding 200 mm/day. The bottom panel shows that although many extreme rainfall events are
23

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Spatial Distribution Precipitation
30
100
45
40
Latitude
-10
1.5
80 50
103 Longitude
120
150
Figure 7: Spatial Distribution ERA5 Annual Precipitation Relative Errors. The relative error is computed as the ratio between absolute bias and station-observed annual precipitation. High values indicate severe misrepresentation of total rainfall.
Temporal Evolution ERA5 Precipitation Interpolation Errors
Spring
Summer
Autumn
Winter
Erowe Metries Meen Iirnor RMSE
Frror (mmetaap)
M pany M Wy
10
Jon
Peb
Mac
Apc
May
Jun
Jul
Oct
Nov
Doo Winter'
Spring
Sumaner exr
Autumn
Maxımum Brror
250
Maxtmum Absolute
WM A MALA py / Mn M mw
Ecror (mm/day)
200
1.50
100
1.90
Spring
Summ err
Autuum
Winter'
Pucceai of Suaieos (%)
Station Distribution Whihemaaiinn ione
mW an Hinhhh
wwmm
MMM
48
My
Jon
Reb
Mac
Age
May
Jun
Pul Date
Aig
Sep
Oct
Nov
Deo
Jan
Figure 8: Temporal Evolution of ERA5 Precipitation Interpolation Errors. Top: daily root mean square error (RMSE) and mean error; Middle: maximum and maximum absolute errors; Bottom: percentage of stations with overestimation underestimation. Shaded regions mark seasonal transitions.

---

underestimated in magnitude, large proportion of stations still experience consistent overestimation throughout the year.
D Origins of Station Heterogeneity and Oversmoothing Biases
Despite the increasing availability of gridded reanalysis products such as ERA5, their direct interpo- lation to station locations often introduces non-trivial biases, partieularly in regions with complex terrain or sparse observational coverage. These systematic errors give rise to two fundamental challenges in station-level precipitation modeling: station heterogeneity and oversmoothing.
Station Heterogeneity. As illustrated in Figures and bilinearly interpolated ERA5 precipitation fields exhibit highly uneven error distributions across space. Some stations experience persistent overestimation, while others are subject to substantial underestimation, particularly during periods of intense rainfall. This spatially inconsistent performance indicates that interpolation biases are inherently location-dependett and cannot be effectively corrected using globally shared model pa- rameters, Importantly this heterogeneity extends beyond spatial variability. As shown in Figure g interpolation accuracy also varies across seasons: errors are generally lower during dry periods, but escalate significantly during the rainy season. This seasonal inconsistency underscores the temporal heterogeneity of the interpolation task, where the statistical properties precipitation and the associ- ated error patterns- -haoenns-hhageeeeee dynamically over time. Therefore, models that rely on static assumptions or uniform temporal behavior are likely underperform in real-world applications. Capturing both spatial and temporal heterogeneity is essential for robust and generalizable interpolation.
Oversmoothing of Extremes. The temporal error profile shown in Figure 8 reveals that, although the overall interpolation error increases during the rainy season, the more critical failure lies in the underestimation of extreme rainfall events. In particular, the maximum error metrics in summer often exceed 200 mm/day, while the corresponding observed values are significantly higher. This pattern indicates that coarse-resolution gridded models tend to suppress sharp gradients and attenuate local extremes through spatial averaging. Such smoothing leads to the loss of physically meaningful high-magna maggitage
relevant for hydrological risk and extreme event forecasting. Consequently, downstream learning models trained on these smoothed inputs may underrepresent rare yet impactful weather patterns, limiting their utility in high-stakes applications.
Together, these findings highlight the empirical origins of the two central challenges tackled in this work. They motivate the design of interpolation frameworks that are not only spatially adaptive but also capable of preserving localized extremes. Subsequent sections build upon these insights to introduce model components that explicitly address heterogeneity and oversmoothing in a unified manner.
E Theoretical Analysis of OU-Based Temporal Encoding
To support the design of the GeoLiquidOU module, we provide a theoretical derivation of its Orn- stein-Uhlenbeck (OU)-inspired update mechanism. This section analyzes how the model integrates historical input over time via exponential smoothing, and how behavior corresponds to first-order low-pass filter in the frequency domain.
E.1 Continuous-Tinn Time OU Dynamics
We begin with the continuous-time OU process, excluding noise terms for clarity:
dh n: [0(x(t)) h(t)),
dt
where ((() is the hidden state, (x((t)) denotes an input-driven target signal, and is > 0 is the decay rate.
Using an integrating factor st and solving the ODE, we obtain:
h(t) h(O)e I' ke- ((t-(())) ((==(s)))((x(s))) ds,
(5)
25

---

which reveals that the current state is a decayed memory of initial state plus an exponentially weighted integration of past inputs. The kernel (t)
acts as memory filter with decay rate K.
E.2 Discrete- Timee Approximation via Euler Method
To implement the update in discrete time with step size At, we apply Buler's method:
1 + 1] = h(m) + At (0 [n] h(n]]
= (1 <x Mm [r],
(6)
where a x&t E (0, 1). This update rule corresponds exactly to an Exponential Weighted Moving Average (EWMA), widely used in both time series filtering and biologically inspired computing.
E.3 Explicit Recursion and Temporal Weighting
Unrolling the recursion across N steps yields:
N
M[]= = (1-)a N
MO]
+
Σ
a(1 N-)-000)
2=0
where the contribution of past inputs decays exponentially. The unnormalized weight for timestep i
is:
W a(1 - a)) a-
(8)
and the normalized weight becomes:
a(1 a) N-1-i (1 - a)
W
Σ5= wj
This shows that the discrete update implicitly applics a form of exponential memory over past values, where smaller a values yield longer memory horizons.
OU Attention Weight Distribution
333-
6.M
10
14
TO
10
5)
[a
0.5
0:25
600
10
59
0.7
3.2
177k
1.99
M00
S00el
Figure 9: Visualization of OU attention weights.
E,4 Frequency-Domain Interpretation: A Low-Pass Filter
We now analyze the above recursion as a linear time-invarian (LTI) system with input [/]]] and output hin). Applying the z-transform yields the transfer function:
02
((x)
(10)
(1 a0)=
26

---

Evaluating on the unit circle = = gives the magnitude response:
H(e H(e²))/=
a
(11)
2(1 a) cos + (1 (1=x)²
This confirms that the system behaves classic first-order low-pass filter. Its key frequency characteristics are:
Zero frequency (w 0):
H H(
=1.
11 a)
This indicates full preservation of low-frequency (DC) components, allowing long-term trends to pass through unchanged.
Highest frequency (w T):
( a)2 2
This shows that figh-frequency signals (rapid temporal variations) are strongly attenuated, effectively denoising short-term fluctuations.
General behavior: The magnitude response (e(r) decreases monotonically with in- creasing w. The parameter controls the cutoff: smaller values of a lead to longer memory and stronger smoothing.
E,5
Interpretation for GeoLiquidOU
This analysis shows that the GeoLiquidOU module dynamically adjusts the temporal sensitivity of each station by learning a site-specific decay factor a = =(p) K, where rpp) a location-awar time constant. This enables the model function learnable, spatially cantally--onditionne low-pass filter, selectively integrating temporal context based on local variability.
Latent-Space Noise Injection and UNet Denoising with Residual Perturbation
Let 2g = F(xo)) E I be the latent-space representation of original interpolated value ag- Under the DDPM framework [18 36), the forward diffusion (noise injection) at step is defined by
24 v 0zz 20 + vI arr
e~ N0, La).
so that
M( 20, (1 a1)) Here, ag (0, 1) denotes the precomputed soise-schedule coefficient.
In the reverse (denoising) step, a UNet model co (zz, f) is trained to predict the injected noise. Denote the prediction error by
8 = E 24. f) -,
which can be modeled zero-mean Gaussian random vector with covariance L3La. The corre- sponding one-step reconstruction in latent space is then
1
2t I
au co(2),)))
20
V1 Ot 8.
Von
Since 8 ~ o( it follows that
20 N 20 1-as of Iz
Because the residual covariance L-as of La is positive definite, the reconstructed latent vector t0 or admits nonzero variance around 2p- In other words, despite denoising, the model retains a controlled stochastic perturbation in latent space, which-when subsequently eecoded-enables recovery or amplification of extreme values beyond the initial interpolated range.
27

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G Cross- Variabl Forecasting: Temperature as Input
To verify the adaptability of our model architecture to alternative meteorological variables, we conduct a parallel experiment using 2-meter temperature (T2M) fields from Pangu-Weather. Weather. While the entire model is retrained from scratch on the T2M dataset, the network structure remains unchanged.
Distribution Model
10
Latitide
00gg
20
Longitude
Figure 10: Spatial Distribution of Absolute Temperature Errors. The annual mean absolute error (MAB) of 2-meter temperature predictions across China using our method DiffLiG.
Spatial Distribution of Model Annual T2M Relative Errors
30
Latiode
35
20
Langit ude
Figure 11: Spatial Distribution of Relative Temperature Errors. The relative error percentage (%) of 2-meter temperature predictions, highlighting spatial performance differences across varied terrain and climate zones.
28

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Annual Mean T2M:: Predicted vs Observed
2,288 stations)
8.0
Predicted T2M (Celsius)
6.0
2.0
Best Fit:
6.86x + 0.12
8.0
2.0
10.0
Observed 12M (Celsius)
Figure 12: Scatter Plot of Predicted vs Observed Annual Mean T2M. This plot shows station- level comparisons between predicted and observed annual average temperatures. A best-fit line (y = 0.86% + 0.12) provided alongside the identity line.
Temperature (T2m) Prediction Performance Analysis
RMSE
ACC
E.56
4373
A311
E:55
6/041
6.0335
0.30
RUICE
6.53
6.50
CAREZ
1.25
Figure 13: Performance Comparison on 2-meter Temperature Prediction. Our method DifILiG, retrained using Pangu-Weather Weather T2M inputs, achieves the best RMSE and ACC scores among baseline methods including MGNN, GCN, and classical interpolation techniques.
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H Sensitivity to Forecast Horizon and Graph Connectivity
To evaluate the robustness and stability of our model across different forecasting and structural configurations, we conduct a series of sensitivity experiments. Specifically, we examine the impact of four critical parameters:
Forecast lead time,
Retrospective window size,
Maximum number of edges from ERAS grid points to stations
Maximum number of inter-station connections.
Unless otherwise specified, the model is trained with the following default configuration: forecast lead time of 1 day, retrospective window size of 5 days, maximum of 4 connections from each ERAS grid point to nearby stations, and up to 3 connections between observation stations. The goal of this analysis is assess the sensitivity of model performance with respect to these parameters, thereby providing insight into the design choices behind spatial and temporal coupling strategies.
H.1 Sensitivity to Forecast Lead Time
Forecast Performance Metrics by Lead Time
5.8
RMSE (mm)
5.6
5.4
4.8
0.740
0.735
ACC
0.730
0.715
Threat Score (T5@2mm)
0.48
0.45
0.42
0.39
0.36
0.33
Lead Time (days)
Figure 14; Impact of Lead Time on Forecast Accuracy.
To assess how prediction horizon affects model performance, we conduct a sensitivity analysis by varying the lead time from to days. As shown in Figure [4] the model retains relatively high accuracy and low RMSE at short-term forecasts. However, performance deteriorates as the prediction window extends, particularly in terms of event detection (TS @ @mm). This suggests that while the
30

---

model can extrapolate coarse patterns over longer horizons, precise localization of extreme events becomes more challenging.
H.2
Impact Backtracking Window Size
Impact of Backtracking Window Size on Forecast Performance
5:12
RMSE (mm)
5.10
5.06
5.04
5.02
7
11
ACC
11
Threat Score (75@2mm)
0.468
0.462
0.456
0.450
0.444
11
Backtracking Window Size (days)
Figure 15: Forecast performance under different temporal backtracking window sizes.
Figure 15 illustrates how the backtracking window iize-i.e.. the number of past days used for station- level temporal encoding- affects forecast performance. As the window length increases, performance steadily improves across all metrics, suggesting that longer temporal context helps capture the heterogeneous evolution of local precipitation patterns. To balance accuracy and computational efficiency, choose 5-day window the default configuration.
H.3 Impact of ERA5-to-Station Edge Connectivity
Figure 16 illustrates the impact of varying the maximum number of ERAS-to-station edges on prediction performance. As the number of edges increases, we observe consistent improvements across RMSE. However, the performance gain plateaus beyond 4 edges, indicating diminishing returns with denser spatial connections. Based on this trend, we select as the default edge limit in model configuration to balance accuracy and computational efficiency.
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Impact of ERAS-Station Edge Connectivity on Forecast Performance
5.5
RMSE (mm))
5.4
5.3
5.1
5.0
ACC
0.735
0.732
0.729
3
Threat Score (T5@2mm)
0.46
0.45
0.44
Maximum Number of Edges from ERAS to Stations
Figure 16: Forecast performance across different ERAS-to-station edge limits.
Impact of Maximum Station-to-Station Connectivity on Forecast Performance
5.14
RMSE (mmm
5.10
5.08
0.740
0.735
ACC
6.725
Threat Score 2mm)
0.485
0.460
6.400
Maximum Number of Station-to-Station Connections
Figure 17: Forecast performance under different inter-station connectivity levels.
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H.4 Impact of Station-to-Station Connectivity
Figure 17 shows the effect of varying the number of station-to-statien connections on forecast connectivity. increasing inter-station edge counts leads to
performance. Unlike the only minor fluctuations RMSE, ACC, and TS@2mm. The performance remains relatively stable across different settings, indicating that the model is less sensitive to this parameter. We therefore fix the maximum number of inter-station edges at for subsequent experiments.
I Implementation and Training Details
I.1 Training Objective and Loss Function
The proposed graph- based forecasting framework is trained in supervised regression setting. The objective is to minimize the discrepancy between the predicted values and ground-truth observations meteorological stations, thereby enabling accurate station-level interpolation.
Let JJ denote the predicted value and Y/ the corresponding ground-truth observation for the i-th sample across time and space. The loss function is defined as the mean squared error (MSE) over all station samples:
N Σ
MMSE
10:
ys1l
(12)
N
where N is the total number of training samples. This loss directly supervises the end-to-end prediction pipeline, encouraging the model to align with observed precipitation patterns across both spatial and temporal dimensions.
1.2 Task Configuration
We formulate the task as a station-level spatial interpolation problem, where the model observes past
Ti days of gridded data predict the station value at the next day (T2 1). The temporal
granularity is set to daily. For each target station, the maximum number of spatial edges is constrained ensure computational efficiency and physical relevance:
Grid-to-station connections: at most 4 ERA5 grid nodes;
Station-to-station connections: at most 3 surrounding stations.
We use 7-year dataset ranging from 2016 to 2022. Specifically, we assign the years 2016-2020 for training, 2021 for validation, and 2022 for final testing. All models are implemented using the PyTorch deep learning framework.
1.3 Training Configuration
The entire training process conducted on single NVIDIA RTX 4090D GPU with 24GB of memory. Each training epoch requires approximately minutes. The total training spans 150 epochs and completes in about 7.5 hours. This setup demonstrates strong computational efficiency and scalability.
1.4 Staged Training Strategy
We progressively activate key modules stabilize training dynamics and ease inter- module adaptation. The training process follows the table
Haversine Distance Computation
To measure the great-cirele distance between two stations the Earth's 's surface, employ the Haversine formula, which accounts for the spherical geometry of the planet.
Let the two geographic locations be denoted as p; (Ai, and P; (Aj, Φ5), where À and represent the longitude and latitude (in degrees), respectively. These coordinates are first converted into radians:
33

---

Table 6: Training Hyperparameters
Parameter
Value
Framework
Py PyTocc
Optimizer
Adam 20
Initial Learning Rate
le-4
Weight Decay
le-4
Batch Size
16
Epochs
150
Time Window
T = 5 (past), T2 = 1 (future)
Grid Edges (max)
4
Station Edges (max)
3
Loss Function
MSE Loss
Dropout
0.1
Activation
LeakyReI.U Dynamic'Tant 43
Training Years
2016-2020
Validation Year
2021
Test Year
2022
Table 7: Staged Training Strategy
Stage
Epoch Range
Activated Modules
Warm-up
0)-20
Basic GNN backbone with fixed topology
GNN Training
20-40
Activate Dynamic Edge Modulator
Forecast Diversity
40-60
Enable Probabilistic Forecast Diffuser
Selector Adaptation
60-80
Introduce Colti-Crittriia Score Selector
Station-we Bias
80-100
Learn station-specific bias correction
Joint Training
100-150
Fine-tune all modules jointly for synergy
Apna
rad di
π
(13)
180
180
We then compute the difference in latitude and longitude:
raa
AA
Aaad
rad
(14)
The Haversine formula estimates the spherical distance dij between the two points as follows:
AΦ
AA
sin"
+ cos(o cas( cos(d)ad) sin"
(15)
2
2
arctan 2 (Va, vl v1
(16)
dæ R
(17)
Here, R 6371 km denotes the average radius of the Earth.
K
Baseline Method Descriptions
To evaluate the effectiveness and generalizability of our proposed framework, we compare against broad of baseline models, categorized into four methodological groups. These baselines cover range of paradigms from classical statistical interpolation to advanced neural forecasting models.
(1) Traditional Interpolation Methods. We include four classic spatial interpolation techniques widely used in geostatistics and meteorology:
Nearest Neighhor assigns each station the value of the closest grid point without smoothing, often leading to abrupt spatial transitions.
34

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Linear Interpolation assumes uniform variation between nearby points, but does not capture spatial correlation terrain effects.
Inverse Distance Weighting (IDW) assigns weights inversely proportional to distance, introducing smooth decay but lacking adaptivity to local density or directional features.
Ordinary Kriging (OK) estimates station values using spatial covariance models (vari- ograms), providing statistically grounded predictions but requiring stationarity assumptions and manual variogram fitting.
While these methods are computationally efficient and interpretable, they rely static spatial assump- tions, lack learnable parameters, and do not account for temporal context or spatial heterogeneity, resulting in degraded performance over complex terrains data-sparse regions.
(2) Graph Neural Network (GNN) Models. GNNs provide a flexible framework for modeling spatial relationships among stations by learning representations through message passing:
GCN M applies localized spectral convolutions on fixed station graphs, but tends to oversmooth signals across the network.
38 introduces attention mechanisms to weigh neighbor importance, improving expressiveness but still relying on static edges.
GAT
GraphSAGE [4] supports inductive learning via sampled aggregation, yet assumes homo- geneous information flow across all regions.
KCN 3 combines kernel-based weighting with graph convolutions to capture geostatistical structure, but uses fixed kernels and non-adaptive connectivity.
Despite their spatial modeling strengths. these GNNs often employ fixed topologies and uniform propagation rules, which limit their ability to adjust to -ation--pecific characteristics and introduce oversmoothing high-resolution prediction tasks.
(3) Physics- and Multimodal Graphs. We also include two architectures that incorporate physical priors or multimodal inputs:
MeshGraphNet BT
learns graph-based simulators aligned with physical PDEs, capturing spatial continuity but requiring large data and stable mesh structures.
MGNN 39] constructs heterogencous graphs combining reanalysis grids and observational stations, enabling multimodal fusion but still relying on predefined graph rules and lacking dynamic edge adaptation.
These models offer better spatial fidelity than standard GNNs but often inherit rigid structures or handcrafted designs, limiting flexibility in highly heterogeneous or evolving forecast settings.
(4) Generative and Transformer-Basee Spatial Models. To assess uncertainty modeling and spatial diversity, we include:
Diffusion Models [8] generate samples via iterative noise reversal, allowing diverse outputs
but often requiring long sampling chains and lacking explicit graph structure.
Vision Transformer (VIT) [1 applies global self-attention over spatial patches, capturing long-range dependencies but prone to global averaging, which may obscure local extremes or abrupt transitions.
These models demonstrate strong generative capacity, yet often lack station-aware structure and may smooth over localized variations, making them less suitable for fine-grained, heterogeneous interpolation tasks.
Together, these baselines form representative and challenging benchmark suite, allowing to assess not only interpolation accuracy but also spatial adaptivity, generalization capacity, and robustness to heterogencity.
35

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L
Evaluation Metrics
We adopt three widely used metrics evaluate interpolation and prediction aceuracy for station-level precipitation: Anomaly Correlation Coefficient (ACC), Root Mean Square Error (RMSE), and Threat Score (TS) under a mm threshold. The formal definitions are provided below.
Given the true values ys and predicted values Di at each location i, we define their respective climatological means To and is as the multi-year averages over reference period for the observed and predicted data, respectively. The anomaly correlation coefficient (ACC) is then computed as:
En Di 0.)
ACC
(18)
(yi 1.)²
(0: -
ACC measures the pattern similarity between predicted and observed anomalies, accounting for inter-location variance.
Root Mean Square Error (RMSE). RMSE evaluates the overall deviation between predictions and ground truth, and defined
RMSE
U4
(19)
where is the total number of evaluated samples. RMSE is sensitive to large errors and commonly used in regression tasks.
Threat Score at 2mm (TS@2mm) To assess the model's ability to detect light to moderate precipitation events, we calculate the Threat Score (TS) under a fixed threshold mm/day:
TP
T 2mm
(20)
TP + FN
FP
where TP (true positives), FN (false negatives), and FP (false positives) are determined based on thresholded binary classification:
TP 2 2 2]. FN Σ 1/u 22 <]
FP E
1lye < < 0. 2 2]
TS reflects the model's event-level accuracy in a binary detection task and is widely used in precipita- tion forecast verification.
M License and Attribution Details
This study uses several existing assets whose licenses and usage terms are properly acknowledged and respected:
Model License. The proposed model will be released under the CC-BY 4.0 license upon publication.
Baselines. All baseline models used in our experiments are implemented based on open- source releases from their original papers. We cite the corresponding works in our main text, and all implementations respect the original licenses provided by their authors.
ERA5 Data. We use the hourly precipitation product from the ERA5 reanaly- sis dataset, provided by European Centre for Medium-Range Weather Forecasts (ECMWF). This dataset is publicly accessible via the [Copernicus Climate Data http:p/s//ddds.iiatt..oeeeeeeeee version released in 2022.
and is licensed under CC-BY 4.0. We use
36

---

The ground truth observation data from the China Meteorological Administration (CMA) is not publicly available. Access to this dataset was granted us through proper institutional authorization.
CMA Observation Data.
No scraped, re-packaged, or third-party datasets outside these sources are used in this study.
N
Visual comparison of spatial interpolation results
Oberservation
Interpolation
MGNN
Area
Area
LIGAP
Area
Area
Diff GifTLiG(ours))
Area
Area
Figure 18: Visual comparison of spatial interpolation results. Predicted daily accumulated precipitation for July 2022.
37


Reviews:
[{"review_id": "ovGWdPPdea", "reviewer": "Reviewer_cW4J", "summary": "This paper proposes DiffLiG, a complex three-component graph neural network for correcting gridded meteorological forecasts to station-level observations. The architecture combines GeoLiquidNet (using Ornstein-Uhlenbeck dynamics), Dynamic Edge Modulator, and Probabilistic Diffusion Selector. While evaluated on Chinese meteorological data, the paper fails to justify why such complexity is needed for what is essentially a spatial interpolation task.", "questions": "Why is graph-based modeling necessary at all? Your ablation study keeps the GCN in all variants. What happens if you remove ALL graph components and use a simple MLP that directly maps from grid points to stations? This fundamental comparison is missing.\nWhy use GCN when recent work shows transformer-based architectures are superior for spatial tasks? Vision Transformers and spatial attention mechanisms have shown better performance on geographic data. Why stick with message-passing GNNs that are known to suffer from oversmoothing?\nWhy not use Neural Operators for this continuous field interpolation? Recent work on Fourier Neural Operators and DeepONet are specifically designed for function approximation between continuous fields. Your grid-to-station task seems perfectly suited for these architectures - why force it into a graph framework?\nHave you considered Implicit Neural Representations (INRs)? Methods like SIREN or Neural Fields can represent continuous spatial functions and naturally interpolate to any coordinate. This would eliminate your complex graph construction entirely. Why wasn't this explored?\nWhy use outdated GCN when Graph Transformers exist? Recent architectures like GraphGPS, SAN, or Graphormer address many limitations of GCNs. Your 2-layer GCN seems particularly primitive given advances in graph architectures.\nWhy separate temporal and spatial modeling? Recent spatiotemporal architectures like STGNNs, DCRNN, or Space-Time Transformers jointly model both dimensions. Your sequential approach (temporal then spatial) seems suboptimal.\nWhy not use Conditional Normalizing Flows for uncertainty? Your diffusion-based ensemble seems overcomplicated compared to flow-based models that can directly model the conditional distribution P(station|grid) with exact likelihood.\nHave you tried Cross-Attention mechanisms? Instead of fixed graph edges, cross-attention between grid and station features would be more flexible and learnable. Why use hard-coded k-nearest neighbors?\nWhat is the computational cost of your three-module architecture? Please provide concrete runtime and memory comparisons with baselines. How can practitioners justify such complexity for a spatial interpolation task?\nWhere is the statistical validation? Your improvements are often less than 5%. Without error bars, confidence intervals, or significance tests, how do we know these aren't within measurement noise? Please provide proper statistical analysis.\nWhy don't you evaluate on extreme events? Your introduction emphasizes extreme precipitation, yet your evaluation uses standard metrics on all data. What is the performance specifically on the top 1% or 5% of rainfall events?\nHow does your method perform with limited historical data? You assume 5 days of historical observations, but many stations have gaps. What happens with only 1-2 days of history, or none at all?", "limitations": "The authors mention some limitations but miss critical ones:\n\nNo analysis of when the method fails\nNo computational cost discussion\nNo consideration of why operational agencies would adopt this over proven methods\nThe geographic limitation to China is downplayed\n\nMore concerning is what the paper doesn't acknowledge: this appears to be a solution in search of a problem.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "Strengths:\n\nThe paper addresses station-level forecast correction, which has practical applications\nIncludes implementation details in appendices\nAttempts to combine multiple deep learning techniques\n\nWeaknesses:\n\nFundamentally flawed experimental design: The ablation study never removes the GCN backbone, making it impossible to determine if graph-based modeling is even beneficial. This is a fatal flaw that undermines the entire contribution\nUnjustified architectural choices: The use of OU dynamics from \"biological inspiration\" for meteorological data is scientifically questionable. No evidence provided that this offers any advantage over standard approaches\nMissing essential baselines: No comparison with simple MLPs (to justify graphs), AGCRN (state-of-the-art adaptive GNN), or operational weather post-processing systems\nNo statistical validation: Despite marginal improvements (~5%), no error bars, confidence intervals, or significance tests provided\n\n\nOverly complex presentation obscures lack of innovation\nCore claims unsupported: \"heterogeneity\" and \"oversmoothing\" presented as major challenges without evidence (other factors may also be challenging)\nContradictory motivations: emphasizes extreme events but never evaluates on them\n\n\nEvaluated only on China, only on precipitation (temperature relegated to appendix)\nNo cross-regional validation, no extreme weather evaluation\nImprovements too marginal to justify adoption\nNo theoretical insights, purely incremental engineering\n\n\nAll components borrowed from existing work without principled integration\nAppears to chase trends (diffusion models) rather than solve problems effectively\nNo novel contributions to understanding spatial interpolation or meteorological processes", "was_revised": true, "final_justification": "After careful consideration of the authors' rebuttal and additional experiments, I am raising my score from 2 to 3.\n\nIssues that were resolved:\n\nThe authors provided comprehensive ablation studies comparing with MLP, Transformer, SIREN, and other architectures, addressing my concern about whether graph-based modeling is necessary\nStatistical validation with error bars and significance tests was added, addressing the lack of statistical rigor\nExtreme event evaluation metrics (TS@20mm, TS@100mm) were included, partially addressing the disconnect between motivation and evaluation\nComputational cost analysis and cross-dataset validation (MADIS) demonstrate practical feasibility\nThe clarification that linear interpolation remains standard practice in meteorological agencies helps justify the practical value\n\nIssues that remain partially unresolved:\n\nThe architecture still appears overly complex for a spatial interpolation task, though the authors argue it's computationally comparable to baselines\nThe biological inspiration (OU dynamics) for meteorological modeling remains scientifically questionable, despite the provided theoretical justification\nCross-regional generalization beyond China and US datasets would strengthen claims of general applicability\n\nWeight of considerations:\n\nThe authors have made substantial efforts to address the core experimental weaknesses. I maintain some skepticism about the architectural complexity. The practical grounding from meteorological agencies and the computational efficiency make this work potentially adoptable.", "rebuttal": "We sincerely thank you for the thoughtful comments. While some concerns were raised about the value of our work, we would like to clarify that our study is in fact grounded in practical needs and solid experimental design.\n\n> Significance of This Work\n\nOur work is motivated by practical forecasting applications that require accurate, station-level precipitation estimates, such as flash flood alerts, and localized hydrological planning. However, **current large-scale models typically produce gridded forecasts**, which do not directly provide rainfall values at station locations.\n\n**Generalization beyond precipitation.** Our method is proposed as a general framework to bridge this gap. Although we primarily evaluate it on precipitation due to its high variability and modeling difficulty, the framework itself is broadly applicable. As demonstrated in Appendix G, our model transfers well to temperature. Due to space limitations, we focus on precipitation in the main text. We have included additional experiments using the MADIS dataset from the United States and evaluated the wind vector variable, as detailed in our response to Reviewer s17Q. You are welcome to refer to that section if interested.\n\n**ERA5 and station data mismatch.** Many forecasting studies use ERA5 as a reference, but it differs significantly from station observations, as shown in *Evaluation of spatial–temporal variation performance of ERA5 precipitation data in China* (Scientific Reports, 2021). We further demonstrate this discrepancy through experiments in Appendix C. As a result, meteorological agencies typically prefer to evaluate forecasts directly against station-level observations.\n\n**Linear interpolation as the mainstream practice.** We consulted experts from the Meteorological Administration, who confirmed that linear interpolation remains the most commonly used method for post-processing gridded forecasts. We include comparisons with this baseline in our paper.\n\n> station heterogeneity and oversmoothing\n\nAs analyzed in Appendix D, station heterogeneity stems from applying uniform interpolation across sites with differing climate conditions, while oversmoothing arises as models tend to predict averaged values to minimize global error.\n\n> GNN as Framework\n\nAt the beginning, we agree that spatial interpolation tasks ideally call for lightweight and simple solutions. In fact, our model is designed with efficiency in mind: **it can be trained on a single 4090D GPU**, and its parameter count and training time are comparable to baseline methods. Our goal from the outset has been to develop a compact model that still delivers superior performance.\n\nIn the early phase of this project, **we explored a range of architectural choices.** We noticed that the MGNN paper had already experimented with pure MLP-based interpolation. For this reason, we only included MGNN in our main comparisons. Beyond that, we also considered FNO and Transformer models. A major challenge with these approaches is determining how many grid points to include for each station. When adopting global modeling, the dense grid and sparse station setup led to vanishing gradient issues during training.  In fact, observations at a station are typically influenced by a few nearby grid cells. This locality makes GNNs a natural fit for the task.\n\n**we also implemented corresponding experiments**. In practice, the GNN structure we adopt is not as complex as it may seem. It simply connects each station to a few nearby grid points. As shown by the parameter counts, we deliberately scaled up the alternative models to match or exceed our own model's size in order to ensure fair comparisons.\n| Model       | RMSE            | ACC             | TS@2mm           | TS@20mm         | TS@100mm         |\n|-------------|------------------|------------------|------------------|------------------|------------------|\n| Linear      | 5.9033           | 0.6667           | 0.4011           | 0.2764           | 0.01558           |\n| MLP         | 6.6446 ± 0.088   | 0.4324 ± 0.013   | 0.2975 ± 0.003   | 0.0723 ± 0.001   | 0.0095 ± 0.001   |\n| Transformer | 6.7028 ± 0.092   | 0.4122 ± 0.011   | 0.2713 ± 0.005   | 0.0728 ± 0.001   | 0.0136 ± 0.001   |\n| SIREN       | 6.3095 ± 0.072   | 0.4967 ± 0.011   | 0.2910 ± 0.003   | 0.0751 ± 0.001   | 0.0084 ± 0.001   |\n\n> Graph Design\n\nWe would like to clarify that **our model is not a simple GCN**, but incorporates multiple structural enhancements. These include a Dynamic Edge Modulator that adaptively controls edge influence, and a multi-head spatial attention mechanism that models asymmetric spatial interactions. we have compared against AGCRN and Graphormer, as you rightly mentioned.\n\n**Sequential modeling with joint fusion.** As for spatiotemporal modeling, we apply temporal and spatial modules sequentially, but their outputs are fused through shared MLP layers to enable joint reasoning. This approach is consistent with many STGNNs, as summarized in *Zhang et al., 2022, Spatio-Temporal Graph Neural Networks for Predictive Learning in Urban Computing: A Survey*.\n\nWe would like to clarify that **our graph edges are not fixed.** They are adaptively modulated through a learnable Dynamic Edge Modulator. In our early experiments, we also explored more flexible structures. However, this led to training instability and vanishing gradient issues, likely due to the sparse and noisy supervision inherent in this interpolation setting. Moreover, introducing fully learnable connectivity would increase the model’s complexity. By using k-nearest neighbor connections as the initial structure and refining them through edge modulation, we achieve a solution that is stable, interpretable, and adaptable for real-world deployment.\n| Model       | RMSE             | ACC              | TS@2mm           | TS@20mm         | TS@100mm         |\n|-------------|------------------|------------------|------------------|------------------|------------------|\n| Graphormer  | 5.3412 ± 0.1213  | 0.7123 ± 0.012   | 0.3872 ± 0.013   | 0.2911 ± 0.002   | 0.01628 ± 0.001   |\n| AGCRN       | 5.5676 ± 0.162   | 0.6587 ± 0.035   | 0.3927 ± 0.012   | 0.2745 ± 0.002   | 0.01423 ± 0.001   |\n\n> GeoLiquid Module\n\nThe GeoLiquid module adapts the temporal memory based on station characteristics, acting as a learnable low-pass filter. We provide its theoretical foundation in **Appendix E**. To further evaluate its effectiveness, we replaced it with DDPM-style embedding, MLP, and attention-based variants for temporal embedding. \n| Method        | RMSE             | ACC         | TS@2mm        | TS@20mm        | TS@100mm        |\n|---------------|------------------|------------------|------------------|------------------|------------------|\n| DiffLiG (ours) | 4.9971 ± 0.102   | 0.7411 ± 0.012   | 0.4637 ± 0.005   | 0.3122 ± 0.001   | 0.02019 ± 0.001   |\n| DiffMlpG       | 5.1011 ± 0.061   | 0.7351 ± 0.006   | 0.4441 ± 0.003   | 0.2911 ± 0.002   | 0.01801 ± 0.001   |\n| DiffAttG       | 5.1284 ± 0.023   | 0.7388 ± 0.011   | 0.4437 ± 0.005   | 0.2933 ± 0.002   | 0.01833 ± 0.001   |\n\n> Historical Observation Data\n\nYou asked how the model performs with limited historical data. We would like to point out that we already conducted a **detailed sensitivity analysis in Appendix H.2.** Additionally, we supplemented an experiment where no historical station data is available.\n| Setting     | RMSE         | ACC       | TS@2mm      | TS@20mm      |  TS@100mm     |\n|-------------|------------------|------------------|------------------|------------------|------------------|\n| No-History  | 5.1342 ± 0.103   | 0.7351 ± 0.011   | 0.4436 ± 0.004   | 0.2855 ± 0.001   | 0.01783 ± 0.001   |\n\n> Oversmoothing and Probabilistic Diffusion Selector\n\nOversmoothing in our task stems mainly from two sources: **the inherent bias in ERA5 data**(see Appendix D), and **the model’s tendency to underestimate heavy precipitation** in pursuit of better global metrics. To evaluate this, we introduced TS@20mm and TS@100mm as targeted indicators for extreme rainfall, and found that all models exhibit low scores under these thresholds. We visualize a representative extreme rainfall event (Appendix N).\n\nWe clarify that **our diffusion module is not as complex as it may seem.** To address your suggestion, we also implemented Conditional Normalizing Flows for uncertainty modeling.\n| Model         | RMSE             | ACC              | TS@2mm           | TS@20mm         | TS@100mm         |\n|---------------|------------------|------------------|------------------|------------------|------------------|\n| LiGAP         | 5.1589 ± 0.096   | 0.7321 ± 0.013   | 0.4559 ± 0.003   | 0.2819 ± 0.001   | 0.01473 ± 0.001   |\n| Flow-LiGAP    | 5.1965 ± 0.0113   | 0.7324 ± 0.021   | 0.4680 ± 0.004   | 0.2765 ± 0.001   | 0.01526 ± 0.001   |\n\n> Table 1: Model Complexity Comparison Table\n\n| Model         | P (#)   | S (MB) | F (M)   | M (MB) | T (s) |\n|---------------|---------|--------|---------|--------|--------|\n| DiffLiG (ours)| 49,361  | 0.19   | 2383.73 | 252.5  | 161    |\n| MLP           |176,257  | 0.67   | 3318.77 | 471.75 | 163    |\n| Transformer   | 84,801  | 0.32   | 3664.15 | 1367.81| 182    |\n| AGCRN         | 74,842  | 0.29   | 6446.77 | 1488.35| 179    |\n| Graphormer    | 70,725  | 0.27   |11170.28 | 4660.00| 182    |\n| DiffMlpG      | 42,163  | 0.16   | 1747.09 | 189.80 | 158    |\n| DiffAttG      | 49,811  | 0.19   | 1979.48 | 215.96 | 173    |\n| Flow-LiGAP    | 84,100  | 0.32   | 4450.80 | 419.50 | 173    |\n\n**Note:**  \n*P: Number of Parameters | S: Model Size | F: Per-sample FLOPs | M: Peak GPU Memory Usage | T: Average Inference Time per Epoch*\n\n**Due to space limitations, more results are provided in our response to Reviewer nrpT, specifically in Table 1 and Table 2.**\n\n**We sincerely thank you for your valuable suggestions! Your final evaluation of our work is of great importance to us!**"}, {"review_id": "4XaUNgiWNE", "reviewer": "Reviewer_hRjM", "summary": "This paper proposes a novel approach for Precipitation Correction based on graph learning. In details, to avoid oversmoothing and fully leverage the rich information provided by the station-level heterogeneity, the authors introduce a GeoLiquidNet which adapts temporal encoding and a GNN to dynamicly learn spatially adaptive connectivity. Additionally, a probabilistic diffusion selector for ensemble forecasts generation is designed to avoid oversmoothing.", "questions": "My main concenrs are as follows:\n1. Compared to DDPM, the novelty of GeoLiquidNet seems limited. And some designs are not well explained.\n2. The clarity of the figures. \n3. How the oversmoothing problem is sovled.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. This paper focuses on meteorological and remote sensing problems, which represent a real-world scenario with significant research value.\n2. The authors conduct extensive experiments to prove the effectiveness of the proposed method.\n\nCons:\n1. Although facing different scenarios, the GeoLiquidNet is essentially a DDPM-like model, merely applied to the meteorological domain to simulate historical evolution. At its core, it still relies on Gaussian noise for perturbation and generative modeling. Why does employing a Markovian-style diffusion process, along with location-conditioned time-step control, allow the model to flexibly adapt to diverse temporal rhythms across stations? Where exactly does the ``flexibility'' of such a Markov process come from? And what is the design rationale behind this time-step control mechanism?\n2. Some symbols are not explained. For example, what do $W_1$ and $W_2$ represent. And some symbols are used more than once. For example, in line 159, $\\kappa$ denotes the trainable dynamic coefficients, however, in line 175, $\\kappa$ denotes the decay factor. The authors should re-check the whole paper for such errors.\n3. The Figure 2 uses an excessive number of fonts and colors, making it difficult to understand the authors' idea. Moreover, some text elements are hard to read due to poor font and color choices. The figure also lacks a legend, and several elements with different meanings are presented using similar graphical forms, which further adds to the confusion.\n4. Why generating more samples could benefit for the oversmoothing problem? Wouldn't a more complex GNN further exacerbate the oversmoothing problem?\n5. In Figure 1, the explanation of Challenge 2 is quite vague, making it difficult to understand how it demonstrates the over-smoothing problem.", "was_revised": true, "final_justification": "The authors' rebuttal has addressed all my concerns. \n\nHowever, due to the figures in the paper need to be revised for better clarity and some mathematical symbols require correction, I'm not entirely confident that the revised version will be significantly easier for readers to understand. In addition, the explanation of over-smoothing is particularly important for understanding the proposed method and the discussion of the problem. I believe the authors should provide more elaboration on this aspect in the main text rather than placing it in the appendix.\n\nTherefore, taking all these factors into account, along with my confidence level in this area, I have decided to raise my rating to 4 instead of 5.", "rebuttal": "We sincerely thank you for taking the time to engage with our work. We hope that our following clarifications will help address the concerns raised and offer a clearer understanding of our design motivations.\n\n> GeoLiquidNet Module\n\nWhile we understand the comparison to DDPM-like models, we would like to clarify that **our method does not follow the generative diffusion paradigm.** The Gaussian noise term in Eq. (1) serves only as a light perturbation to enhance robustness and temporal variability modeling, rather than playing a central role in the learning or prediction process.\n\nOur model is built upon a continuous-time OU process, where the hidden state evolves according to a location-dependent update rule. **The key to its flexibility lies in a time-step controller** that adapts the integration speed based on each station’s geographic characteristics. Stations with rapid precipitation changes receive shorter time constants for quick adaptation, while stable regions use longer memory for smoothing.\n\nIn essence, this mechanism functions as a learnable, location-aware low-pass filter. We provide a detailed theoretical explanation of this behavior in **Appendix E**.\n\nTo help clarify the distinction between our approach and DDPM-style designs, we conducted **a set of ablation studies** where we replaced GeoLiquidNet’s temporal module with several alternatives, including a DDPM-style embedding, a standard MLP-based embedding, and an attention-based embedding.\n| Method        | RMSE             | ACC         | TS@2mm        | TS@20mm        | TS@100mm        |\n|---------------|------------------|------------------|------------------|------------------|------------------|\n| DiffLiG (ours) | 4.9971 ± 0.102   | 0.7411 ± 0.012   | 0.4637 ± 0.005   | 0.3122 ± 0.001   | 0.02019 ± 0.001   |\n| DiffDdpmG      | 5.1230 ± 0.071   | 0.7321 ± 0.013   | 0.4521 ± 0.003   | 0.2894 ± 0.002   | 0.01908 ± 0.001   |\n| DiffMlpG       | 5.1011 ± 0.061   | 0.7351 ± 0.006   | 0.4441 ± 0.003   | 0.2911 ± 0.002   | 0.01801 ± 0.001   |\n| DiffAttG       | 5.1284 ± 0.023   | 0.7388 ± 0.011   | 0.4437 ± 0.005   | 0.2933 ± 0.002   | 0.01833 ± 0.001   |\n\n> Symbol Usage\n\nWe sincerely apologize for the confusion and inconvenience this may have caused you, and we truly appreciate your careful reading and constructive feedback. \n\nWe apologize for the confusion caused by the ambiguous or duplicated use of certain symbols in the manuscript. Specifically, **W₁** and **W₂** refer to two independent linear projection layers implemented in our codebase (DiffLiG/Source/Modules/GNN/liquidnet.py , lines 143 and 162). \n\nRegarding the repeated use of the same symbol for different meanings, we initially intended to distinguish them through typesetting: the former was bolded to indicate vector-valued parameters, while the latter included a subscript to denote a scalar variable. We will revise these symbols thoroughly in the next version to eliminate ambiguity and ensure consistency throughout the manuscript.\n\n> Figure 1\n\n **Figure 1 is not intended to illustrate the underlying mechanism of the over-smoothing problem in detail.** Rather, it serves as a comic-style overview to highlight the two main challenges addressed in our work: station heterogeneity and over-smoothing.\n\nSpecifically for Challenge 2, the upper solid raindrops represent predicted precipitation, while the lower dashed raindrops indicate missed or smoothed-out extremes. The illustration of the Forecast Diffuser & Selector below aims to convey that, in reality, higher rainfall should be retained, whereas under-predicted low rainfall may reflect the failure to capture extremes due to over-smoothing. This visual cue is symbolic rather than analytical, and is not meant to convey precise technical details.\n\nFor a rigorous and systematic discussion, we refer you to Appendix D, where we provide detailed experiments and quantitative analysis.\n\n> Figure 2\n\nThe use of multiple colors and font styles in Figure 2 was originally intended to visually separate different modules in our framework, given its relatively complex structure.\n\nTo briefly clarify the structure shown in Figure 2:\n\n- The upper half of the figure illustrates the full model pipeline, starting from the station-level inputs (left), going through the GeoLiquidNet and Dynamic Edge Modulator, leading to the Probabilistic Forecast Diffuser and Multi-Criteria Score Aggregator (right).\n- The lower half expands on the internal designs of two core components: the Dynamic Edge Modulator (bottom left) and the Probabilistic Forecast Diffuser (bottom right).\n\nThe figure is meant to align closely with the corresponding modules and subsection names in the main text, and their **ordering is kept consistent between the figure and the written descriptions.**\n\nWe truly value your suggestion. In the revised version, we will work to simplify the visual layout, unify the typography, and add a legend to improve readability. \n\n> Origin and Mitigation of the Over-Smoothing Problem\n\n**Sources of oversmoothing.** Oversmoothing in our task stems mainly from two sources: the inherent bias in ERA5 data, which tends to smooth out local extremes (see Appendix D), and the model’s tendency to underestimate heavy precipitation in pursuit of better global metrics. To evaluate this, we introduced TS@20mm and TS@100mm as targeted indicators, and found that all models exhibit low scores under these thresholds, confirming the difficulty of accurately capturing such extremes.\n\n**Diffusion design as the core solution.** Thank you for the thoughtful question. We would like to clarify that the over-smoothing problem in our model is not addressed simply by generating multiple samples. Instead, the core of our solution lies in introducing a diffusion-based mechanism. As we discuss in Appendix F, this design allows the model to move beyond average-case behavior and better capture rare but significant precipitation events. The purpose of generating multiple samples is to enable probabilistic forecasting and to improve the prediction accuracy through ensemble aggregation. To further verify this, we include additional experiments varying the number of generated samples. Notably, even when only one sample is generated, the model still shows clear improvement in capturing heavy rainfall.\n| Samples |   RMSE    | ACC     | TS@2mm    | TS@20mm  | TS@100mm  |\n|-----------|----------------|---------------|----------------|----------------|----------------|\n| 1         | 5.1241 ± 0.101 | 0.7332 ± 0.010 | 0.4611 ± 0.004 | 0.2992 ± 0.001 | 0.01839 ± 0.001 |\n| 3         | 5.0654 ± 0.099 | 0.7386 ± 0.015 | 0.4622 ± 0.003 | 0.3012 ± 0.001 | 0.02016 ± 0.001 |\n| 5         | 4.9971 ± 0.102 | 0.7411 ± 0.012 | 0.4637 ± 0.005 | 0.3122 ± 0.001 | 0.02019 ± 0.001 |\n| 7         | 4.9995 ± 0.131 | 0.7501 ± 0.016 | 0.4651 ± 0.003 | 0.3132 ± 0.002 | 0.01996 ± 0.001 |\n| LiGAP     | 5.1589 ± 0.096 | 0.7321 ± 0.013 | 0.4559 ± 0.003 | 0.2819 ± 0.001 | 0.01473 ± 0.001 |\n\n**Model capacity is not the solution.** To further examine whether the over-smoothing issue could be attributed to model capacity, we constructed a Large_LiGAP variant by significantly increasing the number of parameters in LiGAP. As shown in the table below, this larger model achieves slightly better RMSE and ACC, but it fails to improve TS@20mm and TS@100mm. In fact, its performance on extreme events remains nearly identical to the original LiGAP. However, we do observe that larger models tend to converge in fewer training epochs.\n\n| Model        | RMSE      | ACC      | TS@2mm     | TS@20mm    | TS@100mm   |\n|--------------|----------------|----------------|------------------|------------------|------------------|\n| Large_LiGAP  | 5.1289 ± 0.011  | 0.7371 ± 0.011  | 0.4607 ± 0.009   | 0.2815 ± 0.001   | 0.01470 ± 0.001  |\n\nThese results demonstrate that simply increasing the model capacity does not alleviate the over-smoothing problem. In contrast, DiffLiG significantly improves the TS metrics at high thresholds, confirming that our diffusion-based design plays a key role in mitigating over-smoothing. The detailed parameter comparison can be found in *Table 1*.\n\n\n**Two-fold evaluation of oversmoothing mitigation.** The mitigation of over-smoothing can be evaluated from two complementary perspectives. First, we selected a representative day with widespread extreme precipitation and visualized the station-level predictions. As shown in Appendix N, our method achieves better spatial fidelity in recovering the intensity and variability of heavy rainfall. Second, we added two additional metrics, TS@20mm and TS@100mm. The improvements on both metrics suggest that our approach is effective in alleviating over-smoothing and enhancing the model's capacity to express extreme values. \n\n**We have organized the performance metrics and computational efficiency of the re-evaluated baselines into tables. Due to space limitations, these results are included in the response to Reviewer nrpT, specifically in Table 1 and Table 2.**\n\n> Table 1: Model Complexity Comparison Table\n\n| Model         | P (#)   | S (MB) | F (M)   | M (MB) | T (s) |\n|---------------|---------|--------|---------|--------|--------|\n| DiffLiG (ours)| 49,361  | 0.19   | 2383.73 | 252.5  | 161    |\n| DiffLiG1      | 49,361  | 0.19   | 2229.74 | 611.67 | 159    |\n| DiffLiG3      | 49,361  | 0.19   | 2306.73 | 611.67 | 159    |\n| DiffLiG7      | 49,361  | 0.19   | 2460.71 | 611.67 | 161    |\n| Large_LiGAP   | 369,858 | 1.41   | 22627.45| 2604.42| 221    |\n| DiffDdpmG     | 53,047  | 0.20   | 2201.16 | 212.35 | 153    |\n| DiffMlpG      | 42,163  | 0.16   | 1747.09 | 189.80 | 158    |\n| DiffAttG      | 49,811  | 0.19   | 1979.48 | 215.96 | 173    |\n\n**Note:**  \n*P: Number of Parameters | S: Model Size | F: Per-sample FLOPs | M: Peak GPU Memory Usage | T: Average Inference Time per Epoch*\n\n**We would be glad to further discuss this with you, as your feedback is truly valuable to us.**"}, {"review_id": "ip9L18IBol", "reviewer": "Reviewer_nrpT", "summary": "This paper introduces DiffLiG, a highly innovative and impactful framework that marks a substantial breakthrough in grid-to-station precipitation correction. It is the first work to systematically tackle the dual challenges of station heterogeneity and oversmoothing with a unified, modular architecture, and to provide comprehensive, large-scale experimental validation across multiple mainstream forecasting sources. Beyond methodological novelty, the significance of this work lies in its ability to enable fine-grained, station-level forecasts from coarse-grained global predictions, representing a critical step toward precision meteorology. DiffLiG is not only adaptable to diverse forecast products, but also capable of enhancing their outputs without modifying their internal structure, making it an indispensable post-processing tool for the next generation of AI-based weather forecasting systems. This work fills a clear methodological and practical gap, and has the potential to set a new direction for future research in high-resolution climate modeling.", "questions": "1. Can the authors provide a more explicit evaluation of the model's performance across different seasons, based on the seasonal precipitation error analysis shown in the appendix?\n\n2. Can the authors provide a comparison of the parameter counts or model complexity between the proposed method and the baselines? This would help clarify the trade-off between predictive performance and computational efficiency, especially for deployment in real-world settings.\n\n3. Could the authors clarify how this method relates to existing super-resolution approaches? If higher-resolution inputs become available, would this framework still provide complementary benefits at the station level?\n\n4. In the variable transfer experiment from precipitation to temperature, what components or hyperparameters of the model were adjusted to achieve the reported results?\n\n5. What was the motivation behind including future station observations at time T2 in the problem formulation? Could the authors elaborate on the reasoning or practical scenarios that justify this design choice?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "### Strengths\n\n1.This work addresses a critical and previously underexplored problem in modern weather forecasting: transforming coarse-grained grid outputs into precise, station-level predictions. As the field moves toward high-resolution, localized, and user-relevant forecasts, this contribution is not only timely but strategically essential. \n\n2.While individual model components are grounded in prior work, the authors demonstrate clear and deliberate architectural innovation by tailoring these modules to the unique challenges of grid-to-station correction. This task-driven refinement is thoughtful and well-justified, representing a meaningful form of innovation.\n\n3.The paper is clearly written and well-organized, with figures and narrative consistently aligned around the two central challenges. The analysis is easy to follow, and the results provide concrete evidence that both identified issues are effectively addressed.\n\n4.The experimental evaluation is thorough, with comparisons against diverse baselines across multiple forecast sources, variable transfer tests, and sensitivity analyses. The release of code and training details further supports reproducibility and transparency.\n\n### Weaknesses\n\n1. The paper does not report the parameter size or computational cost of the proposed method compared to baselines, which would help assess its efficiency and practicality.\n\n2. Some parts of the method are described with limited detail, which may affect clarity and reproducibility for readers less familiar with the setting.", "was_revised": true, "final_justification": "The authors have thoroughly addressed my concerns and questions with clear explanations and well-substantiated reasoning. Notably, the study directly tackles the dual challenges of station heterogeneity and oversmoothing, and demonstrates meaningful mitigation in both aspects. Their responses consistently demonstrate the depth and rigor of the work. Taken together, I maintain my support for accepting this paper.", "rebuttal": "Thank you so much for recognizing the value of our work and for your willingness to support its acceptance! We truly appreciate your encouraging feedback and it means a lot to us! To help you better understand our work, we’ve addressed the questions you raised in more detail below. \n\n> Q1:Can the authors provide a more explicit evaluation of the model's performance across different seasons, based on the seasonal precipitation error analysis shown in the appendix?\n\nThank you for this helpful question! In response, we have **conducted an explicit evaluation of our model's performance across different seasons**, following the seasonal precipitation partitioning shown in the appendix. We compared our method with linear interpolation under the same seasonal split and summarized the results in the table provided.\n\nAs the results show, our method consistently outperforms linear interpolation across all seasons in terms of RMSE, accuracy, and threshold-based scores. Notably, during summer when precipitation is most intense and variable, **our model demonstrates a clear improvement in capturing high-magnitude events.** This suggests that it effectively mitigates the over-smoothing issue commonly seen in traditional methods and achieves significantly enhanced predictive performance under challenging conditions.\n\n**Seasonal Evaluation Results**\n| Season  | Method     | RMSE   | ACC    | TS@2mm | TS@20mm | TS@100mm |\n|---------|------------|--------|--------|--------|----------|----------|\n| Spring  | DiffLiG    | 5.3105 | 0.7354 | 0.474  | 0.3618   | 0.02281   |\n|         | Linear     | 6.4889 | 0.6364 | 0.4403 | 0.3191   | 0.02113   |\n| Summer  | DiffLiG    | 9.1056 | 0.6108 | 0.467  | 0.3695   | 0.02749   |\n|         | Linear     |10.2558 | 0.5337 | 0.4126 | 0.3214   | 0.02386   |\n| Autumn  | DiffLiG    | 3.6167 | 0.7845 | 0.4531 | 0.2951   | 0.01760   |\n|         | Linear     | 4.4120 | 0.6857 | 0.3760 | 0.2531   | 0.01244   |\n| Winter  | DiffLiG    | 1.5181 | 0.8931 | 0.4593 | 0.2328   | 0.01339   |\n|         | Linear     | 2.0364 | 0.8178 | 0.3964 | 0.1918   | 0.00887   |\n\n> Q2:Can the authors provide a comparison of the parameter counts or model complexity between the proposed method and the baselines?\n\nTo provide a fair and transparent comparison of model complexity, we evaluated all methods using a unified pipeline that reports the number of trainable parameters, model size, per-sample FLOPs, peak GPU memory usage, and average inference time. These results demonstrate that our method achieves significantly improved performance while introducing only a modest and controllable increase in model complexity. **The detailed comparison of model size, training cost, and inference efficiency is provided in Table 1.** We also re-evaluated the baseline methods through multiple independent runs to ensure fairness and stability. **The results are reported in Table 2.**\n\n> Q3:Could the authors clarify how this method relates to existing super-resolution approaches? If higher-resolution inputs become available, would this framework still provide complementary benefits at the station level?\n\nWhile our method shares the goal of improving spatial detail with super-resolution techniques, **it follows a fundamentally different path**. Traditional super-resolution requires upsampling the entire forecast field, which is computationally demanding. As highlighted in *The Quiet Revolution of Numerical Weather Prediction* (Bauer et al., 2015), increasing spatial resolution often leads to exponential growth in computational cost due to finer grids and smaller time steps. Even AI-based models, though more efficient than traditional NWP, still face a steep rise in training and inference costs as resolution increases. What's more, AI-based methods are typically trained on reanalysis datasets, which are themselves generated by physical models.\n\nIn contrast, **our method is lightweight and targeted.** It directly interpolates gridded forecasts to station locations using a graph-based approach, allowing training on a single NVIDIA 4090D GPU within 6–7 hours. It avoids the need for global field reconstruction while delivering fine-scale accuracy where it matters most.\n\nImportantly, our framework is resolution-agnostic and fully compatible with future high-resolution forecasts. As upstream models improve, **our method can naturally leverage their outputs to further enhance station-level performance** without requiring any changes to their internal structure.\n\n> Q4:In the variable transfer experiment from precipitation to temperature, what components or hyperparameters of the model were adjusted to achieve the reported results?\n\nAs noted in line 662 of the manuscript, *\"the network structure remains unchanged.\"* To assess the robustness of our model across different meteorological variables, **we deliberately kept all model settings and hyperparameters identical** when transferring from precipitation to temperature. This design choice was intended to validate the generalizability of our framework without relying on task-specific tuning.\n\nIn the context of weather prediction, **precipitation is widely regarded as one of the most challenging variables to forecast** due to its high variability and spatiotemporal discontinuity. In contrast, temperature tends to be more stable and easier to predict. This is also why we focus primarily on precipitation in the main experiments.\n\n> Q5:What was the motivation behind including future station observations at time T2 in the problem formulation? Could the authors elaborate on the reasoning or practical scenarios that justify this design choice?\n\nThe inclusion of future time steps (denoted as T2) in our problem formulation is **motivated by a common practical scenario in operational forecasting.** In many real-world applications, we aim to estimate station-level values for future days, for which ground-truth observations are not yet available. However, model-generated forecasts at these future times are already accessible.\n\nOur framework is designed to bridge this gap by interpolating from the gridded model forecasts at time T2 to predict station-level values at the same future time step. This setup reflects a realistic and meaningful use case: making station-level predictions ahead of time using available gridded forecasts, even when the corresponding station observations do not yet exist.\n\n> Q6:Some parts of the method are described with limited detail.\n\n**Due to space limitations** in the main text, we were not able to include all implementation details. Instead, we focused on presenting the key components and novel contributions of our method in a concise manner.\n\n**To ensure clarity and reproducibility, we have released the full codebase.** We believe this provides readers with the necessary resources to fully understand and replicate our approach. We sincerely appreciate your understanding and are open to including more technical details in an extended version or supplementary materials if needed.\n\n> **Table 1: Model Complexity Comparison Table**\n\n| Model         | P (#)   | S (MB) | F (M)   | M (MB) | T (s) |\n|---------------|---------|--------|---------|--------|--------|\n| DiffLiG (ours)| 49,361  | 0.19   | 2383.73 | 252.5  | 161    |\n| LiGAP         | 39,397  | 0.15   | 1737.22 | 212.44 | 154    |\n| GAP           | 33,650  | 0.13   | 1534.93 | 189.76 | 144    |\n| Graph-bias    | 29,506  | 0.11   | 1462.59 | 182.52 | 143    |\n| Graph         | 24,930  | 0.10   | 1462.58 | 182.50 | 143    |\n| GCN           | 29,031  | 0.11   | 1233.01 | 166.07 | 137    |\n| GAT           | 29,287  | 0.11   | 1253.18 | 226.30 | 150    |\n| GraphSAGE     | 33,127  | 0.13   | 1832.93 | 186.42 | 146    |\n| KCN           | 30,520  | 0.12   | 1309.42 | 171.80 | 141    |\n| ViT           | 51,538  | 0.20   | 3972.84 | 2588.59| 326    |\n| Diffusion     | 54,014  | 0.21   | 2340.91 | 3793.07| 280    |\n| MeshGraphNet  | 41,863  | 0.16   | 2234.15 | 223.91 | 156    |\n| MGNN          | 40,741  | 0.16   | 2464.29 | 224.25 | 158    |\n\n**Note:**  \n*P: Number of Parameters | S: Model Size | F: Per-sample FLOPs | M: Peak GPU Memory Usage | T: Average Inference Time per Epoch*\n\nWe conduct ablation experiments to analyze the contribution of each module. LiGAP removes the diffusion module from our full model. GAP further removes the GeoLiquidNet and temporal attention. Graph-bias disables the dynamic edge modulation, and Graph additionally removes the station bias term.\n\n> **Table 2: Performance Comparison of Baselines**\n\n| Model         | RMSE      | ACC      | TS@2mm     | TS@20mm    | TS@100mm   |\n|---------------|----------------|---------------|------------------|------------------|------------------|\n| DiffLiG (ours)| 4.9971 ± 0.102 | 0.7411 ± 0.012 | 0.4637 ± 0.005   | 0.3122 ± 0.001   | 0.02019 ± 0.001  |\n| LiGAP         | 5.1589 ± 0.096 | 0.7321 ± 0.013 | 0.4559 ± 0.003   | 0.2819 ± 0.001   | 0.01473 ± 0.001  |\n| Linear        | 5.9033         | 0.6667         | 0.4011           | 0.2764           | 0.01558          |\n| GAT           | 5.4984 ± 0.095 | 0.6667 ± 0.015 | 0.3202 ± 0.012   | 0.2986 ± 0.001   | 0.0162 ± 0.001   |\n| GCN           | 5.4832 ± 0.055 | 0.6947 ± 0.012 | 0.4139 ± 0.011   | 0.2836 ± 0.002   | 0.01314 ± 0.001  |\n| GraphSAGE     | 5.6421 ± 0.003 | 0.6495 ± 0.009 | 0.3754 ± 0.003   | 0.2648 ± 0.002   | 0.01571 ± 0.001  |\n| KCN           | 5.3897 ± 0.008 | 0.6932 ± 0.012 | 0.4158 ± 0.007   | 0.2862 ± 0.002   | 0.01357 ± 0.002  |\n| MeshGraphNet  | 5.5322 ± 0.011 | 0.6767 ± 0.010 | 0.4374 ± 0.005   | 0.2925 ± 0.001   | 0.01443 ± 0.001  |\n| MGNN          | 5.4428 ± 0.116 | 0.6834 ± 0.013 | 0.3640 ± 0.009   | 0.2611 ± 0.001   | 0.01531 ± 0.001  |\n\n**We sincerely appreciate your recognition of our work! and your further endorsement would be a great encouragement to us.**"}, {"review_id": "mh1oZtGtI0", "reviewer": "Reviewer_s17Q", "summary": "This paper presents a graph-based approach to improve station level weather forecasting by correcting coarse resolution forecasts which are derived from numerical and AI-based models.  The approach includes three main contributions: a station specific temporal modeling, spatially adaptive modulator, and ensemble correction and uncertainty quantification. The performance of the approach beats existing methods in addressing spatial heterogeneity and smoothing.", "questions": "-Could you discuss computational tradeoffs of diffusion vs deterministic baselines?\n-Can this approach incorporate physical constraints?\n-Can this be generalized for multiple time scales?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n-The paper addresses a well-known problem in weather forecasting.\n-The approach is novel, especially the architecture that combines atmospheric dynamics, adaptive constructions, and diffusion.  The design is modular and addresses different parts of the problem.\n-The paper includes comprehensive ablation study. \n-The model has potential to generalize well across weather use cases and other variables such as temperature forecast.\n\nWeaknesses:\n-The paper could be strengthened by using other datasets.  The geographic region is focused on China.  Geographic diversity is needed to demonstrate the method's use for various climate zones.\n-Few details could be better justified (e.g., internals of ensemble approach).\n-Adding other weather variables will strenghen the forecast.\n-Very little information is provided for computational efficiency even though the model is touted as lightweight.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank you for your positive evaluation of our work! Your recognition greatly encourages us and serves as a strong motivation to further refine and advance our research. We also truly appreciate the thoughtful and constructive comments you have provided. In the following response, we address each of your suggestions in detail, aiming to clarify our methodology and design choices, and to better convey the contributions and significance of our work.\n\n> W1&W3: Geographic diversity is needed to demonstrate the method's use for various climate zones. Adding other weather variables will strenghen the forecast.\n\nThank you for this valuable suggestion. To assess the robustness of our method across different climatic regions and weather variables, **we conducted experiments on the MADIS dataset**, which provides rich meteorological observations from over 350 stations in the Northeastern US (2019–2023).\n\nIn this evaluation, **we focused on the Wind Vector variable** and computed the Mean Error (ME) at multiple lead times (in hours). Without changing the model architecture, our method maintained strong performance, consistently outperforming baselines including MLP and MGNN, as shown in the table below.\n\n| Model          | 1    | 2    | 4    | 8    | 16   | 24   | 36   | 48   | Average ME |\n|----------------|------|------|------|------|------|------|------|------|-------------|\n| Linear         | 2.71 | 2.71 | 2.71 | 2.71 | 2.71 | 2.71 | 2.71 | 2.71 | 2.71        |\n| MLP            | 0.38 | 0.47 | 0.54 | 0.59 | 0.62 | 0.90 | 0.92 | 0.92 | 0.72        |\n| MGNN           | 0.37 | 0.45 | 0.53 | 0.57 | 0.59 | 0.59 | 0.60 | 0.62 | 0.54        |\n| DiffLiG (ours) | 0.32 | 0.37 | 0.50 | 0.52 | 0.57 | 0.58 | 0.60 | 0.61 | 0.51    |\n\n> W2: Few details could be better justified (e.g., internals of ensemble approach). \n\nWe sincerely apologize for the limited details in the main text. Due to space constraints, many internal mechanisms had to be omitted or condensed. To address this, **we have released our complete codebase**, and we kindly invite you to explore the implementation for a more thorough understanding. We also plan to include additional explanations in the appendix in future versions. \n\nSpecifically, regarding the **internals of the ensemble approach**, our method consists of two key modules: `diffusion.py`, which implements the **Probabilistic Forecast Diffuser** to generate diverse candidate predictions through repeated sampling in a diffusion-style process; and `selector.py`, which contains the **Multi-Criteria Score Selector** that evaluates the generated samples using spatial consistency, attention consensus, and extremity-aware metrics, and then selects or fuses the optimal output. These modules are located in `DiffLiG/Source/Modules/GNN/`, and we hope they provide further clarity on the ensemble design.\n\n> W4&Q1: Very little information is provided for computational efficiency even though the model is touted as lightweight. Could you discuss computational tradeoffs of diffusion vs deterministic baselines?\n\n **Higher performance with minimal increase in cost.** We thank the reviewer for pointing out the need for a more detailed discussion of computational efficiency. The full training process can be completed in approximately 6–7 hours using a single NVIDIA 4090D GPU, which we consider a relatively small and acceptable computational cost. To address this, we provide quantitative comparisons of the computational costs across methods mentioned in the paper. These are summarized in Table 1. Notably, LiGAP represents our method under a deterministic setting. This small cost leads to noticeably higher forecasting accuracy, especially in capturing heavy precipitation.\n\nTo further illustrate the tradeoff between computational cost and performance in our diffusion-based modeling, we additionally benchmarked the DiffLiG variant with different numbers of generated samples. These results are included in Table 1 and Table 2, where each configuration is denoted as DiffLiG+sample (e.g., DiffLiG1 for one generated sample, etc.).\n\n> Table 1: Model Complexity Comparison Table\n\n| Model         | P (#)   | S (MB) | F (M)   | M (MB) | T (s) |\n|---------------|---------|--------|---------|--------|--------|\n| DiffLiG5 (ours)| 49,361  | 0.19   | 2383.73 | 252.5  | 161    |\n| LiGAP         | 39,397  | 0.15   | 1737.22 | 212.44 | 154    |\n| GAP           | 33,650  | 0.13   | 1534.93 | 189.76 | 144    |\n| Graph-bias    | 29,506  | 0.11   | 1462.59 | 182.52 | 143    |\n| Graph         | 24,930  | 0.10   | 1462.58 | 182.50 | 143    |\n| GCN           | 29,031  | 0.11   | 1233.01 | 166.07 | 137    |\n| GAT           | 29,287  | 0.11   | 1253.18 | 226.30 | 150    |\n| GraphSAGE     | 33,127  | 0.13   | 1832.93 | 186.42 | 146    |\n| KCN           | 30,520  | 0.12   | 1309.42 | 171.80 | 141    |\n| ViT           | 51,538  | 0.20   | 3972.84 | 2588.59| 326    |\n| Diffusion     | 54,014  | 0.21   | 2340.91 | 3793.07| 280    |\n| MeshGraphNet  | 41,863  | 0.16   | 2234.15 | 223.91 | 156    |\n| MGNN          | 40,741  | 0.16   | 2464.29 | 224.25 | 158    |\n| DiffLiG1      | 49,361  | 0.19   | 2229.74 | 611.67 | 159    |\n| DiffLiG3      | 49,361  | 0.19   | 2306.73 | 611.67 | 159    |\n| DiffLiG7      | 49,361  | 0.19   | 2460.71 | 611.67 | 161    |\n\n**Note:**  \n*P: Number of Parameters | S: Model Size | F: Per-sample FLOPs | M: Peak GPU Memory Usage | T: Average Inference Time per Epoch*\n\nWe conduct ablation experiments to analyze the contribution of each module. LiGAP removes the diffusion module from our full model. GAP further removes the GeoLiquidNet and temporal attention. Graph-bias disables the dynamic edge modulation, and Graph additionally removes the station bias term.\n\n> Table 2: Performance Comparison of Baselines\n\n| Model         | RMSE      | ACC      | TS@2mm     | TS@20mm    | TS@100mm   |\n|---------------|----------------|---------------|------------------|------------------|------------------|\n| DiffLiG5 (ours)| 4.9971 ± 0.102 | 0.7411 ± 0.012 | 0.4637 ± 0.005   | 0.3122 ± 0.001   | 0.02019 ± 0.001  |\n| LiGAP         | 5.1589 ± 0.096 | 0.7321 ± 0.013 | 0.4559 ± 0.003   | 0.2819 ± 0.001   | 0.01473 ± 0.001  |\n| Linear        | 5.9033         | 0.6667         | 0.4011           | 0.2764           | 0.01558          |\n| GAT           | 5.4984 ± 0.095 | 0.6667 ± 0.015 | 0.3202 ± 0.012   | 0.2986 ± 0.001   | 0.0162 ± 0.001   |\n| GCN           | 5.4832 ± 0.055 | 0.6947 ± 0.012 | 0.4139 ± 0.011   | 0.2836 ± 0.002   | 0.01314 ± 0.001  |\n| GraphSAGE     | 5.6421 ± 0.003 | 0.6495 ± 0.009 | 0.3754 ± 0.003   | 0.2648 ± 0.002   | 0.01571 ± 0.001  |\n| KCN           | 5.3897 ± 0.008 | 0.6932 ± 0.012 | 0.4158 ± 0.007   | 0.2862 ± 0.002   | 0.01357 ± 0.002  |\n| MeshGraphNet  | 5.5322 ± 0.011 | 0.6767 ± 0.010 | 0.4374 ± 0.005   | 0.2925 ± 0.001   | 0.01443 ± 0.001  |\n| MGNN          | 5.4428 ± 0.116 | 0.6834 ± 0.013 | 0.3640 ± 0.009   | 0.2611 ± 0.001   | 0.01531 ± 0.001  |\n| DiffLiG1      | 5.1241 ± 0.101 | 0.7332 ± 0.010 | 0.4611 ± 0.004 | 0.2992 ± 0.001 | 0.01839 ± 0.001 |\n| DiffLiG3      | 5.0654 ± 0.099 | 0.7386 ± 0.015 | 0.4622 ± 0.003 | 0.3012 ± 0.001 | 0.02016 ± 0.001 |\n| DiffLiG7      | 4.9995 ± 0.131 | 0.7501 ± 0.016 | 0.4651 ± 0.003 | 0.3132 ± 0.002 | 0.01996 ± 0.001 |\n\n> Q2&W3: Can this approach incorporate physical constraints? Adding other weather variables will strenghen the forecast.\n\nThank you for this insightful question! **It coincides well with our own thinking.** During the early stages of this work, we had already taken note of the NeurIPS 2024 paper *“Generalizing Weather Forecast to Fine-grained Temporal Scales via Physics-AI Hybrid Modeling”*, which demonstrates that incorporating multiple physical variables can enhance prediction performance through implicit physical constraints.\n\n**Trade-off between accuracy and complexity.** Motivated by similar insights, we also explored this idea. While using additional variables does improve accuracy, it introduces significant increases in input dimensionality and model complexity. To maintain the lightweight nature of our framework, we drew inspiration from biological systems and introduced a simplified physical prior. Specifically, we designed GeoLiquidNet based on the Ornstein–Uhlenbeck process, which models local temporal dynamics through a compact evolution equation.\n\n**Modularity and extensibility of the design.** Although we currently use only precipitation as input, our model design is modular and flexible. In fact, It can be extended to incorporate additional meteorological variables and explicit physical constraints. \n\n> Q3: Can this be generalized for multiple time scales?\n\n**Yes, our model can be readily generalized to produce forecasts at multiple temporal scales.** In fact, this flexibility is already implemented in our codebase: as shown in /DiffLiG/Source/DiffLiG_Launcher.py, the parameter at line 20 explicitly controls the temporal resolution of the generated samples.\n\nTo further validate this capability, **we provide evaluations under different temporal aggregation settings in Appendix H.1**, where the model demonstrates consistent performance across various time scales.\n\n\n**Once again, we sincerely thank you for your thoughtful and encouraging feedback! We hope that the above clarifications provide a clearer understanding of our method. We would be delighted to engage in further discussion and exchange of ideas with you.**"}]

Author rebuttal:

Meta-review: This paper presents a graph-based approach to improve station-level weather forecasting by correcting coarse-resolution forecasts derived from numerical and AI-based models. The method introduces three main contributions: (1) station-specific temporal modeling, (2) a spatially adaptive modulator, and (3) ensemble correction with uncertainty quantification.
To fully leverage the heterogeneity of station-level data and avoid oversmoothing, the authors propose GeoLiquidNet, which combines temporal encoding with a GNN that dynamically learns spatially adaptive connectivity. In addition, a probabilistic diffusion selector is designed to generate ensemble forecasts while mitigating oversmoothing. The authors have thoroughly addressed reviewer concerns by adding comprehensive ablation studies, providing statistical validation with error bars and significance tests, incorporating extreme-event metrics, including computational cost analysis and cross-dataset validation.
Decision: Accept (poster)