M$^2$Weather: A Benchmark for Joint Multi-Station and Multi-Variable Weather Forecasting

Rongwen Li; Xiao Wang; Mingyang Wang; Hongwu Liu; Changjian Chen; Zhuo Tang; Kenli Li

cs.LGSep 30, 2026

M $^2$ Weather: A Benchmark for Joint Multi-Station and Multi-Variable Weather Forecasting

Authors: Rongwen Li, Xiao Wang, Mingyang Wang, Hongwu Liu, Changjian Chen, Zhuo Tang, Kenli Li

Organizations: Hunan University · China Meteorological Administration

Abstract

Station weather forecasting is fundamentally shaped by both complex spatial dependencies across stations and strong physical coupling among weather variables. However, existing studies often consider these relationships separately and use different datasets and experimental settings, hindering systematic assessment of their individual and joint contributions. In this paper, we introduce $M^2$ Weather, a benchmark for joint multi-station and multi-variable weather forecasting. Through multi-criteria quality control and station stratification, we collect 2,809 high-quality stations with 5 physically coupled weather variables across three spatial scales: France, Europe, and Global. This multi-scale design lets us examine whether conclusions persist from national to global station networks. We also introduce unified training and evaluation protocols to enable fair comparison of different station-variable modeling paradigms. To further examine the benefits of modeling station-variable relationships, we design a lightweight, plug-and-play adapter. With a trained weather forecasting model, this adapter can introduce missing station or variable relationships without retraining the model. This enables fair and efficient investigation of station-variable relationships. Systematic evaluation of 16 representative models shows the benefits of jointly modeling station and variable relationships. Completing missing relationships further reduces MSE for all adapted models on all three datasets. Together, these results identify the complementary information across stations and variables as an important resource for improving station weather forecasting. Our code can be obtained at https://github.com/hnu-vis/M2-Weather.

Figures & tables

Figure 1: Four station–variable forecasting paradigms: (a) SSSV models each series independently; (b) SSMV captures interactions among variables within a station; (c) MSSV captures interactions across stations for one variable; and (d) MSMV models both. All four use temporal history, but differ in the station and variable context available to each forecast.

Figure 2: Overview of the M 2 Weather benchmark and interaction analysis framework. (a) Dataset construction and station stratification based on temporal alignment, quality, provenance, and metadata. (b) Unified protocol for heterogeneous backbones, with model-specific wrappers preserving native interaction structures. (c) Adapter-based analysis on frozen benchmark backbones, with calibration and variable/station interaction corrections fitted by gradient-free ridge regression.

Paradigm	Baselines
SSSV	DLinear ( Zeng et al., 2023 ) , xPatch ( Stitsyuk and Choi, 2025 ) , PatchTST ( Nie et al., 2023 ) , Timer ( Liu et al., 2024b ) , TimeMoE ( Shi et al., 2025 )
SSMV	iTransformer ( Liu et al., 2024a ) , TQNet ( Lin et al., 2025 ) , DUET ( Qiu et al., 2025 ) , Moirai ( Woo et al., 2024 ) , Timer-XL ( Liu et al., 2025 )
MSSV	Corrformer ( Wu et al., 2023 ) , S 2 Transformer ( Chen et al., 2026 ) , CDPNet ( Xu et al., 2025b ) , EasyST ( Tang et al., 2024 ) , STELLA ( Fu et al., 2025 )
MSMV	HiSTGNN ( Ma et al., 2023 )

Table 2: Baseline coverage under the four station–variable modeling paradigms in M 2 Weather.

		France			Europe			Global
Paradigm	Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
SSSV	DLinear	0.4977	0.4605	$-7.49$	0.4481	0.4087	$-8.79$	0.4508	0.4172	$-7.45$
	xPatch	0.3974	0.3642	$-8.36$	0.3509	0.3280	$-6.53$	0.3587	0.3371	$-6.04$
	PatchTST	0.3905	0.3601	$-7.80$	0.3472	0.3266	$-5.92$	0.3540	0.3341	$-5.61$
	Timer	0.5211	0.4445	$-14.70$	0.4392	0.3845	$-12.47$	0.4547	0.3986	$-12.33$
	TimeMoE	0.4354	0.3845	$-11.69$	0.3798	0.3435	$-9.56$	0.3943	0.3559	$-9.73$

Table 3: MSE (

48\!\rightarrow\!72

hours).

\Delta

is the relative change after adaptation; negative values indicate lower error. HiSTGNN is reported without an adapter. Underlining marks the best native backbone; bold marks the best displayed result (ties at four decimals). Per-variable MSE, MAE, and standard deviations are in Appendix C.3 .

Figure 5: MSE by modeling paradigm, averaged over models. Connected markers compare native and adapted predictions; percentages denote reductions in group means. Parentheses indicate group sizes. HiSTGNN is the native MSMV reference, without an adapter.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

Criterion	M 2 -CS	M 2 -RA
Five-year coverage of every variable	$\geq 0.82$	$\geq 0.20$
Five-year mean coverage	–	$\geq 0.65$
Monthly minimum coverage, $P_{10}$	$\geq 0.60$	–
Monthly fourth-highest coverage, $P_{20}$	–	$\geq 0.50$
Monthly mean coverage, $P_{20}$	–	$\geq 0.50$
Monthly maximum five-variable gap, $P_{90}$	$\leq 168$ h	–

Appendix

Table 4: Station-tiering criteria. A dash indicates that the criterion is not used for the corresponding tier.

Code	Source	Present in release
0	Unresolved missing value	No
1	Quality-controlled exact-hour observation	Yes
2	Quality-controlled observation within 15 minutes	Yes
3	Same-station interpolation of a gap $\leq 12$ h	Yes
4	Nearest ERA5 grid cell at the same hour	Yes
5	Same ERA5 grid cell at a nearby hour	No

Appendix

Table 5: Per-value source codes stored in the imputation mask.

Dataset	$T$	$T_{d}$	$WS$	$P$	$WD$	Overall
Global	1.392	1.953	1.578	2.343	5.576	2.568
Europe	2.608	3.216	3.630	11.242	7.499	5.639
France	0.585	0.650	0.735	1.266	1.381	0.923

Appendix

Table 6: Percentage of values completed by either station interpolation or ERA5.

Split	Start	End	Valid windows
Training	2017-01-01 00:00	2020-01-01 14:00	26,175
Validation	2020-01-01 14:00	2020-07-02 04:00	4,263
Test	2020-07-02 04:00	2022-01-01 00:00	13,029

Appendix

Table 7: Shared temporal splits for France, Europe, and Global. Start times are inclusive and end times exclusive; all times are UTC. Window counts use a one-hour stride.

Paradigm / branch	Model	Global	Europe	France
Backbone parameters
SSSV	xPatch	71,076	71,076	71,076
SSSV	PatchTST	455,112	455,112	455,112
SSMV	iTransformer	2,139,976	2,139,976	2,139,976
SSMV	TQNet	596,840	596,840	596,840
MSSV	S 2 Transformer	2,404,936	2,404,936	2,404,936

Appendix

Table 8: Parameter counts across spatial scales. The upper block lists backbone parameters; the lower block lists only additional adapter parameters for the recorded branch configurations.

Paradigm	Model	France	Europe	Global
SSSV	DLinear	$\mathbf{<0.001}$	$\mathbf{<0.001}$	$\mathbf{0.0044}$
	xPatch	$\mathbf{<0.001}$	$\mathbf{<0.001}$	$\mathbf{0.0058}$
	PatchTST	$\mathbf{<0.001}$	$\mathbf{<0.001}$	$\mathbf{<0.001}$
SSMV	iTransformer	$\mathbf{<0.001}$	$\mathbf{<0.001}$	$\mathbf{0.0058}$
	TQNet	$\mathbf{<0.001}$	$\mathbf{0.0069}$	$\mathbf{0.0313}$
	DUET	$\mathbf{0.0043}$	$\mathbf{0.0193}$	$\mathbf{0.0128}$

Appendix

Table 9: Seed-level significance tests for the MSE reductions in Table 3 . Entries are two-sided paired

t

-test

p

-values after Holm correction within each dataset; bold values are significant at

\alpha=0.05

. Each test uses three paired seeds (

df=2

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.3146\!\pm\!0.0003$	$0.2928\!\pm\!0.0000$	$-6.94$	$0.4411\!\pm\!0.0005$	$0.4259\!\pm\!0.0000$	$-3.44$
xPatch	$0.1977\!\pm\!0.0004$	$0.1921\!\pm\!0.0003$	$-2.81$	$0.3371\!\pm\!0.0003$	$0.3342\!\pm\!0.0003$	$-0.89$
PatchTST	$0.1987\!\pm\!0.0005$	$0.1861\!\pm\!0.0005$	$-6.36$	$0.3393\!\pm\!0.0005$	$0.3298\!\pm\!0.0005$	$-2.79$
Timer	$0.3078\!\pm\!0.0000$	$0.2718\!\pm\!0.0000$	$-11.67$	$0.4235\!\pm\!0.0000$	$0.4034\!\pm\!0.0000$	$-4.74$
TimeMoE	$0.2126\!\pm\!0.0000$	$0.1976\!\pm\!0.0000$	$-7.07$	$0.3476\!\pm\!0.0000$	$0.3377\!\pm\!0.0000$	$-2.83$

Appendix

Table 10: France — Air temperature. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.8359\!\pm\!0.0003$	$0.7586\!\pm\!0.0000$	$-9.25$	$0.6746\!\pm\!0.0014$	$0.6506\!\pm\!0.0000$	$-3.56$
xPatch	$0.7956\!\pm\!0.0006$	$0.7054\!\pm\!0.0003$	$-11.34$	$0.6394\!\pm\!0.0003$	$0.6171\!\pm\!0.0003$	$-3.49$
PatchTST	$0.7811\!\pm\!0.0002$	$0.7029\!\pm\!0.0001$	$-10.01$	$0.6374\!\pm\!0.0004$	$0.6165\!\pm\!0.0001$	$-3.28$
Timer	$0.9058\!\pm\!0.0000$	$0.7490\!\pm\!0.0000$	$-17.31$	$0.6968\!\pm\!0.0000$	$0.6434\!\pm\!0.0000$	$-7.68$
TimeMoE	$0.8696\!\pm\!0.0000$	$0.7362\!\pm\!0.0000$	$-15.34$	$0.6694\!\pm\!0.0000$	$0.6328\!\pm\!0.0000$	$-5.47$

Appendix

Table 11: France — Wind speed. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.7059\!\pm\!0.0007$	$0.6835\!\pm\!0.0000$	$-3.16$	$0.6742\!\pm\!0.0020$	$0.6734\!\pm\!0.0000$	$-0.12$
xPatch	$0.4959\!\pm\!0.0007$	$0.4671\!\pm\!0.0006$	$-5.82$	$0.5305\!\pm\!0.0005$	$0.5316\!\pm\!0.0005$	$+0.22$
PatchTST	$0.4882\!\pm\!0.0005$	$0.4657\!\pm\!0.0005$	$-4.61$	$0.5346\!\pm\!0.0001$	$0.5311\!\pm\!0.0003$	$-0.67$
Timer	$0.7680\!\pm\!0.0000$	$0.6644\!\pm\!0.0000$	$-13.50$	$0.6638\!\pm\!0.0000$	$0.6551\!\pm\!0.0000$	$-1.31$
TimeMoE	$0.5592\!\pm\!0.0000$	$0.5141\!\pm\!0.0000$	$-8.07$	$0.5644\!\pm\!0.0000$	$0.5635\!\pm\!0.0000$	$-0.16$

Appendix

Table 12: France — Relative humidity. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.1346\!\pm\!0.0004$	$0.1069\!\pm\!0.0000$	$-20.53$	$0.2715\!\pm\!0.0013$	$0.2379\!\pm\!0.0000$	$-12.38$
xPatch	$0.1005\!\pm\!0.0004$	$0.0922\!\pm\!0.0003$	$-8.25$	$0.2211\!\pm\!0.0006$	$0.2141\!\pm\!0.0006$	$-3.16$
PatchTST	$0.0941\!\pm\!0.0006$	$0.0856\!\pm\!0.0005$	$-8.99$	$0.2121\!\pm\!0.0009$	$0.2044\!\pm\!0.0008$	$-3.64$
Timer	$0.1030\!\pm\!0.0000$	$0.0929\!\pm\!0.0000$	$-9.73$	$0.2222\!\pm\!0.0000$	$0.2133\!\pm\!0.0000$	$-4.03$
TimeMoE	$0.1002\!\pm\!0.0000$	$0.0901\!\pm\!0.0000$	$-10.06$	$0.2174\!\pm\!0.0000$	$0.2084\!\pm\!0.0000$	$-4.13$

Appendix

Table 13: France — Station pressure. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.2270\!\pm\!0.0003$	$0.2107\!\pm\!0.0000$	$-7.15$	$0.3684\!\pm\!0.0004$	$0.3536\!\pm\!0.0000$	$-4.03$
xPatch	$0.1481\!\pm\!0.0002$	$0.1445\!\pm\!0.0001$	$-2.41$	$0.2861\!\pm\!0.0002$	$0.2836\!\pm\!0.0002$	$-0.89$
PatchTST	$0.1505\!\pm\!0.0006$	$0.1432\!\pm\!0.0006$	$-4.84$	$0.2898\!\pm\!0.0008$	$0.2836\!\pm\!0.0008$	$-2.16$
Timer	$0.2039\!\pm\!0.0000$	$0.1859\!\pm\!0.0000$	$-8.83$	$0.3386\!\pm\!0.0000$	$0.3263\!\pm\!0.0000$	$-3.63$
TimeMoE	$0.1561\!\pm\!0.0000$	$0.1482\!\pm\!0.0000$	$-5.09$	$0.2917\!\pm\!0.0000$	$0.2858\!\pm\!0.0000$	$-2.03$

Appendix

Table 14: Europe — Air temperature. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.7531\!\pm\!0.0004$	$0.6782\!\pm\!0.0000$	$-9.94$	$0.6265\!\pm\!0.0014$	$0.6085\!\pm\!0.0000$	$-2.87$
xPatch	$0.6890\!\pm\!0.0004$	$0.6318\!\pm\!0.0002$	$-8.29$	$0.5836\!\pm\!0.0002$	$0.5780\!\pm\!0.0002$	$-0.96$
PatchTST	$0.6775\!\pm\!0.0008$	$0.6272\!\pm\!0.0001$	$-7.42$	$0.5830\!\pm\!0.0002$	$0.5762\!\pm\!0.0001$	$-1.18$
Timer	$0.7625\!\pm\!0.0000$	$0.6617\!\pm\!0.0000$	$-13.23$	$0.6238\!\pm\!0.0000$	$0.5979\!\pm\!0.0000$	$-4.16$
TimeMoE	$0.7422\!\pm\!0.0000$	$0.6554\!\pm\!0.0000$	$-11.68$	$0.6048\!\pm\!0.0000$	$0.5905\!\pm\!0.0000$	$-2.35$

Appendix

Table 15: Europe — Wind speed. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.7719\!\pm\!0.0006$	$0.7191\!\pm\!0.0000$	$-6.85$	$0.6820\!\pm\!0.0019$	$0.6794\!\pm\!0.0000$	$-0.39$
xPatch	$0.5427\!\pm\!0.0001$	$0.5119\!\pm\!0.0002$	$-5.68$	$0.5439\!\pm\!0.0003$	$0.5416\!\pm\!0.0003$	$-0.43$
PatchTST	$0.5371\!\pm\!0.0015$	$0.5131\!\pm\!0.0016$	$-4.46$	$0.5502\!\pm\!0.0015$	$0.5499\!\pm\!0.0011$	$-0.05$
Timer	$0.7665\!\pm\!0.0000$	$0.6671\!\pm\!0.0000$	$-12.98$	$0.6468\!\pm\!0.0000$	$0.6440\!\pm\!0.0000$	$-0.43$
TimeMoE	$0.5974\!\pm\!0.0000$	$0.5476\!\pm\!0.0000$	$-8.33$	$0.5686\!\pm\!0.0000$	$0.5617\!\pm\!0.0000$	$-1.22$

Appendix

Table 16: Europe — Relative humidity. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.0404\!\pm\!0.0004$	$0.0268\!\pm\!0.0000$	$-33.59$	$0.1505\!\pm\!0.0017$	$0.1128\!\pm\!0.0000$	$-25.07$
xPatch	$0.0238\!\pm\!0.0001$	$0.0236\!\pm\!0.0001$	$-0.64$	$0.1023\!\pm\!0.0003$	$0.1022\!\pm\!0.0003$	$-0.06$
PatchTST	$0.0236\!\pm\!0.0001$	$0.0229\!\pm\!0.0001$	$-2.99$	$0.1017\!\pm\!0.0003$	$0.1003\!\pm\!0.0003$	$-1.34$
Timer	$0.0240\!\pm\!0.0000$	$0.0232\!\pm\!0.0000$	$-3.04$	$0.1016\!\pm\!0.0000$	$0.1001\!\pm\!0.0000$	$-1.48$
TimeMoE	$0.0234\!\pm\!0.0000$	$0.0227\!\pm\!0.0000$	$-3.23$	$0.0986\!\pm\!0.0000$	$0.0972\!\pm\!0.0000$	$-1.47$

Appendix

Table 17: Europe — Station pressure. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.1837\!\pm\!0.0008$	$0.1685\!\pm\!0.0011$	$-8.25$	$0.3285\!\pm\!0.0009$	$0.3124\!\pm\!0.0012$	$-4.88$
xPatch	$0.1186\!\pm\!0.0010$	$0.1160\!\pm\!0.0011$	$-2.25$	$0.2507\!\pm\!0.0015$	$0.2495\!\pm\!0.0015$	$-0.46$
PatchTST	$0.1188\!\pm\!0.0000$	$0.1139\!\pm\!0.0001$	$-4.11$	$0.2514\!\pm\!0.0001$	$0.2478\!\pm\!0.0001$	$-1.45$
Timer	$0.1628\!\pm\!0.0000$	$0.1500\!\pm\!0.0000$	$-7.88$	$0.2979\!\pm\!0.0000$	$0.2895\!\pm\!0.0000$	$-2.80$
TimeMoE	$0.1258\!\pm\!0.0000$	$0.1208\!\pm\!0.0000$	$-3.95$	$0.2562\!\pm\!0.0000$	$0.2533\!\pm\!0.0000$	$-1.12$

Appendix

Table 18: Global — Air temperature. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.8466\!\pm\!0.0053$	$0.7770\!\pm\!0.0010$	$-8.22$	$0.6877\!\pm\!0.0024$	$0.6675\!\pm\!0.0006$	$-2.94$
xPatch	$0.7833\!\pm\!0.0041$	$0.7216\!\pm\!0.0003$	$-7.87$	$0.6473\!\pm\!0.0013$	$0.6348\!\pm\!0.0004$	$-1.94$
PatchTST	$0.7767\!\pm\!0.0005$	$0.7179\!\pm\!0.0002$	$-7.57$	$0.6473\!\pm\!0.0002$	$0.6333\!\pm\!0.0001$	$-2.17$
Timer	$0.8832\!\pm\!0.0000$	$0.7617\!\pm\!0.0000$	$-13.76$	$0.6968\!\pm\!0.0000$	$0.6576\!\pm\!0.0000$	$-5.62$
TimeMoE	$0.8533\!\pm\!0.0000$	$0.7477\!\pm\!0.0000$	$-12.38$	$0.6747\!\pm\!0.0000$	$0.6476\!\pm\!0.0000$	$-4.01$

Appendix

Table 19: Global — Wind speed. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.7290\!\pm\!0.0071$	$0.6969\!\pm\!0.0040$	$-4.41$	$0.6677\!\pm\!0.0031$	$0.6673\!\pm\!0.0023$	$-0.05$
xPatch	$0.5100\!\pm\!0.0042$	$0.4878\!\pm\!0.0035$	$-4.36$	$0.5264\!\pm\!0.0023$	$0.5214\!\pm\!0.0024$	$-0.95$
PatchTST	$0.4980\!\pm\!0.0006$	$0.4829\!\pm\!0.0008$	$-3.03$	$0.5285\!\pm\!0.0009$	$0.5261\!\pm\!0.0006$	$-0.46$
Timer	$0.7494\!\pm\!0.0000$	$0.6601\!\pm\!0.0000$	$-11.92$	$0.6430\!\pm\!0.0000$	$0.6371\!\pm\!0.0000$	$-0.91$
TimeMoE	$0.5753\!\pm\!0.0000$	$0.5332\!\pm\!0.0000$	$-7.32$	$0.5577\!\pm\!0.0000$	$0.5611\!\pm\!0.0000$	$+0.61$

Appendix

Table 20: Global — Relative humidity. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

	MSE $\downarrow$			MAE $\downarrow$
Model	Base	+Adapter	$\Delta$ (%)	Base	+Adapter	$\Delta$ (%)
DLinear	$0.0438\!\pm\!0.0013$	$0.0263\!\pm\!0.0000$	$-39.83$	$0.1603\!\pm\!0.0019$	$0.1163\!\pm\!0.0001$	$-27.50$
xPatch	$0.0230\!\pm\!0.0004$	$0.0229\!\pm\!0.0004$	$-0.59$	$0.1043\!\pm\!0.0012$	$0.1043\!\pm\!0.0012$	$-0.08$
PatchTST	$0.0224\!\pm\!0.0001$	$0.0218\!\pm\!0.0001$	$-2.71$	$0.1026\!\pm\!0.0004$	$0.1015\!\pm\!0.0004$	$-1.05$
Timer	$0.0232\!\pm\!0.0000$	$0.0225\!\pm\!0.0000$	$-2.85$	$0.1046\!\pm\!0.0000$	$0.1033\!\pm\!0.0000$	$-1.22$
TimeMoE	$0.0227\!\pm\!0.0000$	$0.0220\!\pm\!0.0000$	$-3.06$	$0.1021\!\pm\!0.0000$	$0.1009\!\pm\!0.0000$	$-1.16$

Appendix

Table 21: Global — Station pressure. MSE and MAE (mean

\pm

sample standard deviation).

\Delta

(%) is signed relative change; negative is better.

Figure 6: Individual-model counterpart of Figure 5 . Open circles denote native predictions, filled circles denote adapted predictions, and squares mark the unadapted HiSTGNN. Thin rules separate SSSV, SSMV, MSSV, and MSMV; lower MSE is better.

France
Model	Base	Interaction only	Full
DLinear	$0.4977\,\pm\,0.0004$	$0.4800\,\pm\,0.0004$	$\mathbf{0.4605}\,\pm\,0.0000$
xPatch	$0.3974\,\pm\,0.0004$	$0.3858\,\pm\,0.0003$	$\mathbf{0.3642}\,\pm\,0.0002$
PatchTST	$0.3905\,\pm\,0.0004$	$0.3724\,\pm\,0.0004$	$\mathbf{0.3601}\,\pm\,0.0003$
Timer	$0.5211\,\pm\,0.0000$	$0.4704\,\pm\,0.0000$	$\mathbf{0.4445}\,\pm\,0.0000$
TimeMoE	$0.4354\,\pm\,0.0000$	$0.4028\,\pm\,0.0000$	$\mathbf{0.3845}\,\pm\,0.0000$

Appendix

Table 22: Calibration ablation on France and Global. MSE is reported as mean

\pm

sample standard deviation across seeds 2024–2026. Lower is better. Full includes calibration and the same interaction branches. HiSTGNN is excluded.

Backbone	Residual source	MSE $\downarrow$	$\Delta$ (%)
PatchTST	Base	$0.4030\pm 0.0001$	—
	In-sample full	$\mathbf{0.3703}\pm 0.0001$	$-8.11$
	In-sample matched	$0.3790\pm 0.0001$	$-5.94$
	Held-out	$0.3716\pm 0.0001$	$-7.79$
iTransformer	Base	$0.3813\pm 0.0005$	—
	In-sample full	$\mathbf{0.3525}\pm 0.0003$	$-7.54$

Appendix

Table 23: Sensitivity to residual fitting sources on France. MSE is reported as mean

\pm

sample standard deviation over three seeds.

\Delta

is the mean per-seed percentage change relative to the shared Base; negative values indicate improvement. Bold marks the lowest mean MSE per backbone.

		France		Europe
Paradigm	Backbone	Time (min)	Memory (GiB)	Time (min)	Memory (GiB)
SSSV	xPatch	11.68	0.05	3.02	0.23
	PatchTST	9.95	1.37	1.96	7.56
SSMV	iTransformer	6.48	0.77	2.83	4.18
	DUET	13.81	0.71	3.15	3.56
MSSV	Corrformer	40.67	26.31	30.93	30.03

Appendix

Table 24: Backbone training cost at batch size 2. Time is minutes per training epoch; memory is peak GPU allocation in GiB.

	France			Europe
Frozen backbone	Fit time (min)	Memory (GiB)	Time ratio (%)	Fit time (min)	Memory (GiB)	Time ratio (%)
xPatch	2.77	0.023	23.7	0.64	0.093	21.1
PatchTST	2.33	0.880	23.5	0.80	4.550	40.6
iTransformer	2.93	0.700	45.3	0.62	2.560	22.0
DUET	3.32	0.690	24.1	0.71	2.510	22.6
Corrformer	16.01	28.720	39.4	12.16	21.350	39.3

Appendix

Table 25: Complete fitting cost of the full dual-branch adapter with a frozen backbone.

T_{\mathrm{fit}}/T_{\mathrm{epoch}}

compares total adapter fitting time with one training epoch of the same backbone.

Figure 7: MSE versus look-back window on France with a fixed 72-hour forecast horizon. Each curve reports the mean MSE across repeated runs for a native backbone without an adapter. HiSTGNN maintains the lowest MSE across all four windows.

Figure 8: Sensitivity of the station-interaction adapter to the KNN neighborhood size

K

. Each curve shows the three-run mean MSE averaged over four variables; lower is better.

Figure 9: Air temperature forecasts on Europe. Station ITMU0016052; forecast start 2020-12-31 00:00 UTC. Each panel compares a native backbone with its full adapter over the same 72-hour window following 48 hours of history. Adapted forecasts better follow the sustained warming trend.

Figure 10: Wind speed forecasts on Europe. Station NOI0000ENFG; forecast start 2021-12-06 18:00 UTC. Each panel compares a native backbone with its full adapter over the same 72-hour window following 48 hours of history. Adapted forecasts capture the increase in wind speed, while rapid fluctuations remain difficult to reproduce.

Figure 11: Relative humidity forecasts on Europe. Station ROM00015317; forecast start 2021-01-25 06:00 UTC. Each panel compares a native backbone with its full adapter over the same 72-hour window following 48 hours of history. Adapted forecasts better track the humidity decline, with residual differences in timing and magnitude.

Figure 12: Station pressure forecasts on Europe. Station AUM00011126; forecast start 2021-05-27 19:00 UTC. Each panel compares a native backbone with its full adapter over the same 72-hour window following 48 hours of history. Most adapted forecasts follow the observed pressure level and gradual variation more closely; STELLA retains pronounced fluctuations.

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StationPDE: Station-Oriented Surface PDE Learning for Multi-Station Multivariate Weather Forecasting

Xiao Wang, Changjian Chen, Rongwen Li, Hongwu Liu+2

Hunan University.

Weather Prediction Spatio-Temporal Forecasting Wind Partial Differential Equations Digital Elevation Models

Open paper AskPDF arXiv

Multi-station multivariate weather forecasting aims to forecast future weather variables at multiple weather stations from historical surface observations. Existing station forecasting models learn statistical dependencies among discrete stations, but lack explicit physical evolution. Meanwhile, PDE-based weather models provide interpretable physical dynamics, yet require continuous fields and upper-air variables unavailable in surface station data. To bridge this gap, we propose StationPDE, a station-oriented surface PDE learning model. StationPDE constructs a terrain-aware continuous surface field from discrete station observations and decomposes its physical evolution into surface wind transport and upper-air inference. Surface wind transport explicitly evolves observable weather variables, while upper-air inference uses learnable horizontal diffusion to approximate the missing influence of unavailable upper-air variables. A parallel data-driven diffusion branch captures complementary motion patterns, and an adaptive router integrates the two forecasts for station-level multivariate forecasting. Experiments on Weather2K and MeteoNet show that StationPDE consistently outperforms state-of-the-art baselines, reducing MSE by about

9.6\%

on average compared with the strongest baseline.

Abstract

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