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Conditional Local Convolution for Spatio-temporal Meteorological Forecasting

Haitao Lin, Zhangyang Gao, Yongjie Xu, Lirong Wu, Ling Li, Stan. Z. Li

arXiv:2101.01000v3cs.LG

TL;DR

Meteorological forecasting must model irregular spherical sensor locations alongside complex spatial and temporal dynamics. The paper proposes a smoothness-based conditional local graph convolution and embeds it in a recurrent network; the authors report state-of-the-art weather-forecasting performance and identify over-smooth learned patterns and training instability as issues for future work.

  • Problem

    Meteorological forecasting involves irregularly distributed spherical sensors and complex spatial and temporal dependencies, while the paper notes less research attention to meteorology than traffic forecasting.

    Source: §1 · p. 1, §4.4 · p. 5

  • Method

    The method uses a shared conditional local convolution based on smooth location-characterized patterns, with a recurrent network for temporal modeling.

    Source: §1 · p. 1, §4.3 · p. 4, §6 · p. 8

  • Results

    The authors report state-of-the-art weather-forecasting performance and improved empirical performance, without supplying a numerical comparison in these passages.

    Source: §1 · p. 1, §6 · p. 8

  • Takeaways & Limitations

    The proposed convolution is designed to imitate local meteorological flows on the sphere while accounting for uneven node distribution.

    Source: §4.3 · p. 5, §6 · p. 8

    The authors describe the convolution as inapplicable to traffic forecasting because smoothness of local traffic patterns and continuity of orientation may not hold for irregular road networks.

    Source: §4.3 · p. 4, §4.4 · p. 5

Abstract

from arXiv · show

Spatio-temporal forecasting is challenging attributing to the high nonlinearity in temporal dynamics as well as complex location-characterized patterns in spatial domains, especially in fields like weather forecasting. Graph convolutions are usually used for modeling the spatial dependency in meteorology to handle the irregular distribution of sensors' spatial location. In this work, a novel graph-based convolution for imitating the meteorological flows is proposed to capture the local spatial patterns. Based on the assumption of smoothness of location-characterized patterns, we propose conditional local convolution whose shared kernel on nodes' local space is approximated by feedforward networks, with local representations of coordinate obtained by horizon maps into cylindrical-tangent space as its input. The established united standard of local coordinate system preserves the orientation on geography. We further propose the distance and orientation scaling terms to reduce the impacts of irregular spatial distribution. The convolution is embedded in a Recurrent Neural Network architecture to model the temporal dynamics, leading to the Conditional Local Convolution Recurrent Network (CLCRN). Our model is evaluated on real-world weather benchmark datasets, achieving state-of-the-art performance with obvious improvements. We conduct further analysis on local pattern visualization, model's framework choice, advantages of horizon maps and etc.

1. Introduction

Meteorological forecasting combines irregular spherical sensor locations with complex spatial and temporal dependencies, while meteorology has received less attention than traffic forecasting. The paper proposes a location-conditioned graph convolution for local patterns and evaluates its recurrent model alongside further analyses.

1. Introduction

  • Irregular sensor locations on a spherical surface and strong spatial and temporal dependencies make meteorological forecasting difficult for conventional grid-based CNNs.The introduction also notes that landforms can produce distinct flow patterns and that extreme climate incidents such as El Niño can make prediction non-stationary.Source: §1 · p. 1
  • The proposed graph convolution uses a location-characterized kernel to capture and imitate local meteorological spatial patterns.The convolution is performed in each node's local space, considering distance and relative orientation, under an assumption of smoothness in location-characterized patterns.Source: §1 · p. 1
  • The authors report that their spatio-temporal model achieves state-of-the-art performance in weather forecasting tasks.They also analyze learned local-pattern visualization, framework choice, local-space and map choice, and ablations.Source: §1 · pp. 1–2

2. Related Work

Prior work uses graph methods to handle irregular spatial signals and forecast spatio-temporal data, especially traffic. The paper distinguishes its approach by imposing smoothness on learned local meteorological patterns.

Spherical signal processing

  • Graph-based methods process spherical or irregularly distributed signals, while some mesh-based spherical convolutions impose properties such as equivariance but require mesh-grid data and can be computationally costly.The paper also places its method among graph-based approaches to spherical signals.Source: §2 · p. 2

Spatio-temporal graph neural networks

  • Prior spatio-temporal graph models aggregate, diffuse, or spectrally filter neighborhood information, and traffic models include graph attention and adaptive local patterns.The authors distinguish their method as adaptively learned local patterns constrained by smoothness, intended to capture meteorological flow.Source: §2 · p. 2

3. Background

The background formulates forecasting as mapping historical signals on a spherical graph to future signals. It introduces local spaces and graph convolution as the basis for defining how neighborhood messages are aggregated.

Problem setting

  • The forecasting setup represents correlated signals on the sphere as a graph and learns to map historical observations to future signals.For datasets without an adjacency matrix, the paper constructs one using spherical-distance K-nearest neighbors.Source: §3 · p. 2

Graph convolution neural networks

  • A graph convolution updates each node by aggregating messages from its neighborhood using a convolution kernel.The kernel assigns weights to neighboring nodes; the paper relates larger weights to neighbors with greater meteorological influence, such as heat flow from the southeast toward the northwest.Source: §3 · p. 2

Sphere manifold

  • A local space is a Euclidean space centered at a point and homeomorphic to its surrounding local region, with the tangent space given as an example.An isometric map can send spherical neighbors into that local space while preserving their distance from the center.Source: §3 · p. 3
  • On the sphere, the shortest intrinsic distance is the great-circle distance, which the paper uses for K-nearest-neighbor graph construction.The logarithmic map is described as an isometric way to map a spherical neighbor into the center's tangent space; local coordinates are then set using a transform mapping and orthogonal bases.Source: §3 · p. 3

4. Proposed Method

The method performs graph convolution in each node’s local space, using horizon maps and a shared, center-conditioned kernel to model meteorological patterns. It combines distance and orientation scaling for irregularly distributed neighbors, then embeds the convolution in a recurrent forecasting architecture.

4.1. Local convolution on sphere

  • Local convolution maps graph neighbors into each center node’s local space and restricts convolution to a finite neighborhood using a local kernel.Source: §4.1 · p. 3

4.2. Orientation-preserving local regions

  • Horizon maps place neighbors in a cylindrical-tangent local space that preserves geographic orientation and spherical distance; the transforms cannot be applied at the poles.Source: §4.2 · p. 4

4.3. Conditional local convolution

  • The conditional local kernel uses local coordinates and the center node as inputs, with smoothness across nearby centers assumed and distance and angle scales addressing irregular neighbor distributions.Source: §4.3 · pp. 4–5

4.4. Inapplicability in traffic forecasting

  • The convolution is inapplicable to traffic forecasting because nearby traffic patterns may differ sharply and road-constrained flows lack meteorology’s continuous spatial-flow setting.Source: §4.4 · p. 5

4.5. Temporal dynamics modeling

  • CLCRN replaces recurrent-cell fully connected layers with graph convolutions and uses a sequence-to-sequence encoder-decoder for multi-step forecasting.Source: §4.5 · p. 6

5. Experiments

Across weather-forecasting evaluations, CLCRN improves on recurrent baselines, while learned humidity kernels show smoothness and directionality but are over-smoothed. The experiments also compare recurrent and attention-based frameworks, local-coordinate maps, and kernel components.

5.1. Experiment setting

  • The evaluation uses four WeatherBench weather tasks, with 2048 Earth-sphere nodes and 12 time steps each for input and forecast; MAE, RMSE, and MAPE are the metrics.Results for compared methods are averaged over five random-seed experiments; Table 1 contrasts their spatial and temporal modules and whether their spatial kernels are learnable or continuous-domain.Source: §5.1 · p. 6, Table 1 · p. 6

5.2 Performance comparison

  • CLCRN further improves recurrent methods in weather prediction, while recurrent methods generally outperform attention-based methods except on Temperature, where MSTGCN works well.The comparison uses MAE and RMSE at a 12-hour forecast horizon; Figure 3 plots the top three methods by MAE across forecasting lengths.Source: §5.2 · p. 6, Table 2 · p. 7, Figure 3 · p. 7

5.3. Visualization of local patterns

  • Humidity kernels vary smoothly across uniformly sampled centers and show directionality, with north-west and south-east nodes impacting centers most, but the patterns are over-smoothed.The authors identify over-smoothness as a future research issue.Source: §5.3 · p. 7, Figure 4 · p. 7

5.4. Framework choice: CNN or RNN?

  • CLCRN outperforms CLCSTN in all evaluations, with a larger gap in short-term than long-term prediction; the authors associate this pattern with attention fitting low-frequency signals better than short-term fluctuations.Figure 5 illustrates predictions during steep slopes and drastic fluctuations, and Figure 6 shows the learning curves; recurrent-model training instability remains a future research issue.Source: §5.4 · pp. 7–8, Table 3 · p. 7, Figure 5 · p. 8

5.5. Advantages of horizon maps

  • Table 4 compares MAE and RMSE at a 12-hour forecast horizon for logarithmic maps in tangent space and horizon maps in cylindrical-tangent space.The supplied experimental text identifies the comparison but does not report its outcome.Source: Table 4 · p. 8

5.6. Ablation study

  • The Distance scaling term is least important in the reported ablation: MLP + Angle performs almost the same as MLP + Angle + Distance, while Distance alone usually performs worst.The kernel ablation decomposes the kernel into MLP, distance-scaling, and angle-scaling terms; hyperparameter experiments also examine neighbor number K and layer count.Source: §5.6 · p. 8, Table 5 · p. 8, Figure 7 · p. 8

6. Conclusion

The conclusion presents a conditional local convolution for modeling meteorological patterns on the sphere and describes the local-space construction supporting it.

6. Conclusion

  • The proposed convolution uses an MLP kernel and reweighting terms based on continuous relative positions to handle unevenly distributed nodes.It assumes smoothness of location-characterized patterns.Source: §6 · p. 8

A1: Glossary of notations

  • The notation glossary identifies local-space coordinates, neighborhood representations, and the horizon map used to express spherical locations.It also defines the geodesic distance and local coordinate representations used in the method.Source: Table 6 · p. 10, A1: Glossary of notations · p. 10

A2: Formal Definition of Local Space

  • A point’s local space is defined as a Euclidean space homeomorphic to its centered local region.The formal construction uses a homeomorphism from a spherical ball of radius π.Source: A2: Formal Definition of Local Space · p. 10

B1: Proof of Proposition 2

  • The proof concludes that the horizon map is an isometric mapping from spherical nodes into the centered cylindrical-tangent space.The proof first establishes that mapped vectors lie in that space.Source: B1: Proof of Proposition 2 · p. 10

B2: Fast implementation and transforms on poles

The appendix gives a fast coordinate transform for dense spherical sampling and specifies pole handling, then describes the GRU-based temporal dynamics using local convolution.

B2: Fast implementation and transforms on poles

  • For dense spherical sampling, a fast transform can replace the local coordinate transform and horizon map.Its longitude difference is adjusted by ±2π when it falls outside [−π, π]; the resulting mapping preserves graticule orientation but not distance and is reported as competitive.Source: B2: Fast implementation and transforms… · p. 10
  • At either pole, neighbors’ first local coordinate is zero, while their second coordinate is negative at the North pole and positive at the South pole.The method uses the negative relative positions of the pole in each neighbor’s local space.Source: B2: Fast implementation and transforms… · p. 10

C: Temporal Dynamics Modeling

  • The temporal model uses GRU units, with convolution on each node’s local space and the kernel specified by Eq. 15.The passage identifies reset and update gates and describes each unit’s input and hidden state.Source: C: Temporal Dynamics Modeling · p. 11

D1: Experiment protocol and details

The experiment appendix reports training settings, identifies the dataset-statistics table, and defines evaluation metrics while noting a MAPE caveat.

D1: Experiment protocol and details

  • Experiments use five seeds, early stopping on validation data, batch size 32, and one NVIDIA V100 GPU per run.The learning rate starts at 0.01 and decays by a factor of 0.05 every 10 epochs during the first 50 epochs; reported results are means and standard deviations across seeds.Source: D1: Experiment protocol and details · p. 11
  • Table 7 presents dataset statistics for the four Weatherbench datasets, but the supplied passage does not give the table values.The appendix states that further dataset details are in that table.Source: Table 7 · p. 11, D1: Experiment protocol and details · p. 11
  • MAE, RMSE, and MAPE are computed from predictions and ground truth, but MAPE is unstable because its denominator can be zero or extremely small.For Cloud cover, the minimum remains extremely small, making MAPE extremely large.Source: D1: Experiment protocol and details · p. 11

D2: Overall method comparison

The comparison section describes hour-wise metric comparisons and explains why Temperature results converge as forecast horizons grow. Its visualization reports better CLCRN predictions than CLCSTN on datasets with sharper changes.

D2: Overall method comparison

  • Figure 11 compares hour-wise MAE, RMSE, and MAPE for the overall top-three methods on each dataset.The supplied text identifies the comparison scope but does not provide the plotted values or winners.Source: D2: Overall method comparison · p. 11
  • On Temperature, model performance appears to converge at longer horizons because smooth predictions suit its gentle fluctuations, while fitting fluctuations can incur large errors.The text attributes this pattern to longer time scales benefiting smooth outputs and hurting models that try to fit fluctuations.Source: D2: Overall method comparison · p. 11

D3: More visualization of prediction

  • On datasets with significant, drastic changes, CLCRN predictions are more accurate than CLCSTN’s, while CLCSTN remains competitive on gently varying Temperature.The passage describes these patterns in the prediction visualizations.Source: D3: More visualization of prediction · p. 12

D4: Sensitivity of hyperparameters

The sensitivity analysis recommends moderate model capacity: wider hidden representations often improve performance but cost more, while deeper models increase parameters and overfitting risk. Increasing the neighbor count can help until distant, irrelevant messages are aggregated.

Layer number

Neighbor Number

Hidden Units

  • Increasing hidden units generally improves performance, but raises computational time and space costs; the authors recommend 64 to 96.The reported trend is attributed to increased model expressivity.Source: D4: Sensitivity of hyperparameters · p. 13

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