PING: physics informed graph neural network for advection-diffusion processes with applications to solar power forecasting

Jelena Simeunović; Baptiste Schubnel; Pierre-Jean Alet; Pascal Frossard; Rafael E. Carrillo

cs.LGOct 13, 2026

PING: physics informed graph neural network for advection-diffusion processes with applications to solar power forecasting

Authors: Jelena Simeunović, Baptiste Schubnel, Pierre-Jean Alet, Pascal Frossard, Rafael E. Carrillo

Organizations: CSEM, Rue de Jaquet-Droz 1, 2002 Neuchâtel, Switzerland · Department of Electrical Engineering, École Polytechnique Fédérale de Lausanne (EPFL), Rte Cantonale, 1015 Lausanne, Switzerland.

Abstract

Advection-diffusion plays an essential role in transport phenomena such as atmospheric and cloud dynamics. Improved forecasting of cloud motion, in turn, enhances solar power production forecasts and facilitates the integration of additional photovoltaic (PV) sources into the power grid. However, both classical numerical models and emerging deep learning approaches face challenges when applied to spatially irregular sensor networks such as those formed by weather stations. In this work, we adopt a graph signal processing perspective and model advection-diffusion particles as signals on a graph to capture the dynamics of these processes. We introduce a Physics-Informed Graph Neural Network (PING) for forecasting future advection-diffusion particle concentrations on irregular grids. The model incorporates a discretized advection-diffusion equation as a soft constraint in the loss function to guide learning toward physically consistent solutions. This model is on par with state of the art models on synthetic and real transport phenomena datasets, including sea surface temperature and cloud index datasets. Furthermore, the PING model is successfully coupled with models for forecasting PV power production. This hybrid approach leverages data fusion from ground-based sensors at PV stations and satellite images with cloud concentration to improve the accuracy of intra-day PV power production.

Figures & tables

Figure 1: PING model with three main parts: flow estimation, flow attention (edge estimation) and flow processor (flow aggregation). The flow estimation part has as input past concentration values $\bf{C}$ and outputs past velocity and acceleration field estimates. The flow attention takes as inputs the estimated past velocity field $\bf{v}$ and acceleration fields $\bf{a_{cc}}$ as well as the embedded input concentrations $\bf{C_{emb}}$ . The Flow processor predicts the future concentration values $\bf{\hat{C}}$ using the edge values from the second block. When used for PV power prediction, the predicted concentration (cloud concentration in that case) and future clear sky values $\bf{Y}$ are fed to the decoder of the PV power forecasting model.

Figure 2: The flow estimator block of the PING model.

Panel of Figure 3 — Figure 3: The spatial distribution of the subsampling patterns. a) Subsampling set $\mathcal{S}_{1}$ . b) Subsampling set $\mathcal{S}_{2}$ . c) Subsampling set $\mathcal{S}_{3}$ . d) Subsampling set $\mathcal{S}_{4}$ .

Steps ahead		1	2	3	4	5	6
dataset	model	1	2	3	4	5	6
Advection- diffusion dataset	GCLSTM	0.44	0.87	1.46	2.19	3.08	4.28
	GCLSTMmlp	0.79	1.18	2.02	2.97	3.97	4.86
	PING	1.37	1.51	1.85	2.37	2.9	3.4
Advection dataset	GCLSTM	1.51	2.55	3.69	4.87	6.18	7.73
	GCLSTMmlp	1.67	2.51	3.67	4.79	6.09	7.28

Table 1: NRMSE of advection and advection-diffusion datasets on regular grid

Steps ahead		1	2	3	4	5	6
dataset	model	1	2	3	4	5	6
Cloud index dataset	GCLSTM	12.83	18.56	22.03	24.27	25.97	27.26
	GCLSTMmlp	13.39	19.78	23.24	25.44	26.9	28.07
	FlowCNN	13.49	18.55	21.58	23.71	25.3	26.61
	PING	12.21	17.42	20.93	23.47	25.29	26.57
SST dataset	GCLSTM	1.67	2.29	2.89	3.47	4.01	4.48

Table 2: NRMSE of cloud concentration index and sea surface temperature datasets on regular grid

Steps ahead		1	2	3	4	5	6
model	Graph instance	1	2	3	4	5	6
GCLSTM	S1	1.96	2.73	4.06	5.62	7.18	8.42
GCLSTMmlp		1.96	3.1	4.19	5.83	7.38	8.64
PING		3.11	3.97	4.65	5.61	6.74	7.76
GCLSTM	S2	1.92	3.22	4.69	5.9	7.23	8.48
GCLSTMmlp		2.58	3.53	4.63	5.44	6.2	7.2

Table 3: NRMSE of advection datasets on graph instances

S_{1},S_{2},S_{3},S_{4}

Steps ahead		1	2	3	4	5	6
model	Graph instance	1	2	3	4	5	6
GCLSTM	S1	12.52	18.07	21.29	23.46	25.07	26.34
GCLSTMmlp		13.12	19.1	22.57	24.69	26.22	27.38
PING		12.64	17.56	20.83	23.13	24.9	26.12
GCLSTM	S2	12.55	18.01	21.35	23.58	25.24	26.66
GCLSTMmlp		16.45	19.52	23.12	25.38	26.86	27.9

Table 4: NRMSE of cloud concentration index datasets on irregular grid

Steps ahead		1	2	3	4	5	6
model	Graph instance	1	2	3	4	5	6
GCLSTM	S1	1.67	2.29	2.89	3.45	4.01	4.52
GCLSTMmlp		1.77	2.34	2.94	3.51	4.02	4.54
PING		1.88	2.44	2.92	3.37	3.9	4.31
GCLSTM	S2	1.69	2.29	2.88	3.43	3.98	4.51
GCLSTMmlp		1.95	2.52	3.1	3.66	4.2	4.56

Table 5: NRMSE of sea surface temperature datasets on irregular grid

Steps ahead
	1	2	3	4	5	6
$\lambda_{1}=0.15,\ \lambda_{2}=0.05$	0.0321	0.0389	0.0404	0.0480	0.0548	0.0619
$\lambda_{1}=0.20,\ \lambda_{2}=0.05$	0.0250	0.0318	0.0360	0.0399	0.0466	0.0548
$\lambda_{1}=0.15,\ \lambda_{2}=0.10$	0.0299	0.0352	0.0430	0.0478	0.0574	0.0666

Table 6: NRMSE sensitivity analysis of the physics-informed loss weights on the advection–diffusion dataset under irregular sampling.

Dataset	Model	$\phi(rad)$
Advection dataset	PING	0.44
Advection dataset	Optical flow	1.16
Advection-diffusion dataset	PING	0.92
Advection-diffusion dataset	Optical flow	1.10

Table 7: Accuracy of velocity estimation on regular grid for advection and advection-diffusion datasets (PING vs. Optical flows)

Figure 4: Example of velocities estimation on a regular grid, for the advection only dataset. The similarity between the ground truth velocity direction and the estimated one is color-coded: yellow represents the perfect alignment between the vectors and darker colors indicate larger deviations. The arrows represent the direction of the velocity vector, black being the ground truth and red the estimated one.

Figure 5: Example of velocities estimation on an irregular grid, for the advection only dataset. The similarity between the ground truth velocity direction and the estimated one is color-coded: yellow represents the perfect alignment between the vectors and darker colors indicate larger deviations. The arrows represent the direction of the velocity vector, black being the ground truth and red the estimated one.

Figure 6: Combination of PING and GCLSTM for PV power forecasting. PING takes as inputs past cloud cover index values

C

. The encoder of GCLSTM takes as inputs cloud cover index

\bf C

values, historical clear-sky values

\bf Y

and historical observations from PV power plants

\bf P

. The output from PING, the encoder state and future clear-sky values are passed to the decoder.

Figure 7: Evolution of the NRMSE between GCLSTM+PING, GCLSTMmlp and GCLSTM models for six hours ahead for PV power generation, with hourly resolution. The solid line shows the median value among all nodes.

Figure 8: NRMSE evolution of GCLSTM, CloudMove, GCLSTM+PING and persistence model over the forecasting horizon of 6 hours ahead in steps of 15 minutes.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure 9: Examples of historical cloud concentration, ground truth and forecasted signals across six forecasting horizon values for a specific node on irregular grid. The results are shown for PING and GCLSTM models.

Figure 10: Ground truth and prediction for sixth hour ahead forecasts during variable days, where each forecast was initialized always six hour before its verification time.

Panel of Figure 11 — Figure 11: Examples of six hours ahead forecasts initialized at 13:00 and16:00 for the PV power prediction dataset. The forecasted signals and the past signal values are shown for PING+GCLSTM and GCLSTM alone.

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When Does Advection-Aware Graph Nowcasting Help? A Controlled Study of Distributed Solar Ramp Forecasting with a Self-Supervised Cloud-Motion Estimator

Phillip Jiang

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Solar Power Forecasting Spatiotemporal GNNs Graph Neural Networks Optical Flow

Open paper AskPDF arXiv

Short-term forecasting of cloud-induced power ramps across a network of distributed photovoltaic (PV) or irradiance sensors is a recognised pain point for grid operators. A natural idea is to make the graph neural network (GNN) advection-aware: connect each site to the sites upwind of it, with edge time-lags set by the cloud-motion vector (CMV), so that a ramp is propagated forward before it physically arrives. Using a controlled synthetic testbed with a known wind field, we show that (i) with a realistic cross-correlation CMV estimate, an explicit advection graph does not beat a plain static or learned-adjacency spatiotemporal GNN; (ii) roughly half of the benefit available from a perfect CMV comes simply from providing an accurate motion vector as an input feature, not from graph structure; and (iii) advection helps only when the advective displacement over the forecast horizon, v*H, fits inside the sensor network. Motivated by (ii), we introduce a small self-supervised cloud-motion estimator -- a position-aware encoder trained only on a multi-lag optical-flow reconstruction objective with an annealed kernel -- that recovers the true wind vector to 2-4 degrees median angular error, 2-4x better than the classical cross-correlation method across every wind regime. Freezing this estimator and feeding its vector to the forecaster closes about 60% of the oracle-CMV RMSE gap at moderate wind (8-15% RMSE reduction over no advection), with no external wind data. We also report a negative result for a spatially-coherent probabilistic head. All claims are established on a single synthetic simulator; we discuss why real-network validation is the necessary next step and outline it.

Abstract

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