Multi-Horizon Graph Neural Network Emulator

• Multi-horizon graph neural network emulators are learned surrogates that model dynamical systems on graphs and predict future states over multiple time horizons.
• They integrate operator learning with residual and autoregressive strategies, employing architectures like DeepGraphONet, GraphINO, and EGCN to handle diverse forecasting regimes.
• These emulators leverage graph message passing and dynamic adjacency learning to enhance spatial correlation, enable zero-shot transfer, and mitigate error accumulation.

A multi-horizon graph neural network emulator is a learned surrogate that represents a dynamical system on a graph and predicts future states at multiple lead times from observed history, current state, exogenous forcings, or scenario parameters. In the literature, this designation spans several closely related formulations: continuous-time operator-learning models that answer arbitrary time queries within a forecast window, such as DeepGraphONet (Sun et al., 2022); horizon-conditioned residual predictors trained on a discrete set of lead times, such as the long-range Pine Island Glacier emulator (Liu et al., 28 May 2026); and one-step graph simulators that are rolled out autoregressively over many steps, as in graph-based space weather forecasting (Holmberg et al., 23 Sep 2025) and finite-element ice-dynamics emulation (Koo et al., 2024). The common aim is to couple graph-based spatial inductive bias with multi-step forecasting so that future trajectories can be generated more efficiently, and often more flexibly, than by full numerical simulation or fixed-grid neural surrogates.

1. Problem formulation

The canonical setting treats a networked dynamical system on a fixed graph $G=(V,E)$ whose state at time $t$ is a graph signal $X(t)\in\mathbb{R}^{d\times |V|}$. In DeepGraphONet, the dynamics are written as an unknown initial value problem,

$$\frac{d}{dt}\,\mathbf{x}(t) = f(\mathbf{x}(t); G),\quad \mathbf{x}(t_0) = \mathbf{x}_0,$$

with solution operator

$$\mathcal{F}(\mathbf{x}(t_0);G)(t) = \mathbf{x}(t_0) + \int_{t_0}^{t} f(\mathbf{x}(\beta);G)\,d\beta.$$

The forecasting task is to use a finite history over a sub-graph $S\subseteq G$ to predict future trajectories over a continuous multi-horizon window $\tau\in[0,h]$ (Sun et al., 2022).

Other works instantiate the same idea on different graph types. In seasonal forecasting, GraphINO models oceanic grid cells as graph nodes and predicts the Oceanic Niño Index at lead months $n=1,2,3,4,5,6,9,12,23$ from a window of past monthly fields (Cachay et al., 2021). In mesh-native geophysical emulation, finite-element or mesh graphs are built directly from triangular or quadrilateral discretizations, and the targets are nodewise physical states such as ice thickness, horizontal velocities, or plasma variables (Koo et al., 2024). In multivariate time-series forecasting, the graph may instead connect sensors, variables, or products, and the output is a direct vector of future horizons rather than a continuous-time trajectory (Yang et al., 2023).

A central distinction is whether the emulator maps to an entire future window in one evaluation or reaches that window by repeated single-step updates. This distinction governs training, stability, error accumulation, and the interpretation of “multi-horizon,” which is not uniform across the literature.

2. Core architectural patterns

A recurrent architectural motif is the fusion of graph message passing with a temporal or operator-learning module. DeepGraphONet combines a GraphSAGE-style branch network with a DeepONet trunk. The branch encodes historical node values, and in its resolution-independent form also ingests sensor timestamps; the trunk takes a continuous query time $\tau$ and produces a basis vector $\varphi(\tau)$. For node $t$0, prediction is

$t$1

where $t$2 are nodewise branch coefficients produced by the GNN (Sun et al., 2022).

A second pattern is the residual state-transition model. The horizon-aware Pine Island Glacier emulator predicts state increments rather than absolute future states,

$t$3

using a shared five-layer GCN backbone and separate output branches for velocity and thickness (Liu et al., 28 May 2026). This residual formulation is explicitly motivated by stability over long horizons.

A third pattern is encode–process–decode message passing on static or hierarchical meshes. In graph-based space weather forecasting, two consecutive states are encoded, processed by Interaction Networks or Propagation Networks on simple, multiscale, or hierarchical meshes, and decoded into residual field updates (Holmberg et al., 23 Sep 2025). In EGCN-based finite-element ice emulation, equivariant message passing couples node states with geometry so that predictions preserve equivariance to the translation and rotation of graphs (Koo et al., 2024).

Graph construction is likewise heterogeneous. GraphINO learns a static directed adjacency matrix by a dedicated graph structure learning module optimized jointly with the forecasting loss (Cachay et al., 2021). ESG learns a series of adjacency matrices recurrently at each layer and time scale, so that inter-variable dependencies can be both evolutionary and multi-scale (Ye et al., 2022). MKH-Net combines an explicit static graph, an implicit learned hypergraph, and a dual hypergraph, then fuses the resulting spatial experts by a mixture-of-experts gate (Sakhinana et al., 2024).

This suggests that “multi-horizon graph neural network emulator” denotes a family of graph-based surrogate formulations rather than a single architectural template.

3. Forecasting regimes across horizons

The term “multi-horizon” covers at least three distinct operational regimes.

The first is direct continuous-time querying. DeepGraphONet produces direct multi-step predictions inside a horizon by querying any $t$4 without forward dependency, so predictions at different future times are parallel and non-autoregressive within the horizon (Sun et al., 2022).

The second is direct discrete multi-output prediction. GEANN augments an encoder–decoder forecaster so that the decoder outputs a vector of future predictions over horizons $t$5 in one shot (Yang et al., 2023). The horizon-aware geophysical emulator similarly trains one model on a predefined horizon set such as $t$6 and then performs a coarse-to-fine rollout that fills distant times with larger jumps before using shorter jumps for refinement (Liu et al., 28 May 2026).

The third is autoregressive rollout. Graph-based neural space weather forecasting is trained single-step with teacher forcing and then iterated at inference so that

$t$7

with boundary-forced nodes replaced at each step (Holmberg et al., 23 Sep 2025). EGCN-based ice-sheet emulation likewise performs sequential rollout across transient sequences, with the caveat of possible error accumulation (Koo et al., 2024).

A common misconception is that multi-horizon forecasting always implies simultaneous prediction of all lead times by a single head. GraphINO provides a counterexample: it achieves multi-horizon forecasting by training a single model per lead month $t$8, and the paper explicitly leaves simultaneous multi-output heads as future work (Cachay et al., 2021).

4. Graph structure, invariance, and transferability

Graph representation is not merely bookkeeping; it determines what is meant by spatial correlation, locality, and transfer.

DeepGraphONet targets sub-graph forecasting and is designed to be resolution-independent: the branch ingests both values and sensor locations over the history window, and the learned operator is approximately invariant to sampling resolution. It also uses an inductive GNN branch, enabling zero-shot transfer to an unseen connected sub-graph $t$9 without fine-tuning, provided that node and edge feature semantics remain consistent and normalization matches training (Sun et al., 2022).

GraphINO emphasizes learned long-range connectivity. Its adjacency is static with respect to time, globally sparsified by keeping the largest $X(t)\in\mathbb{R}^{d\times |V|}$0 edge weights, with $X(t)\in\mathbb{R}^{d\times |V|}$1 in experiments, and is intended to encode teleconnections rather than short-lived local couplings (Cachay et al., 2021). By contrast, ESG explicitly models adjacency as dynamic and scale-dependent, using recurrently evolved node embeddings to decode a different adjacency sequence at each layer and segment (Ye et al., 2022).

In physical-system emulation, mesh-native modeling is central. Ice-sheet emulators build graphs directly from finite-element connectivity, avoiding rasterization and preserving refined resolution in fast-flow regions (Koo et al., 2024). EGCN strengthens this by enforcing equivariance to translation and rotation of graphs, which the paper identifies as especially important for thickness prediction near fast ice streams (Koo et al., 2024).

Hypergraph constructions broaden the notion of relation beyond pairwise edges. MKH-Net learns a dynamic hypergraph, extracts explicit subgraph patches from domain knowledge, transforms the explicit graph into a dual hypergraph, and fuses all three views; this is used to combine domain-specific knowledge with implicit relational structure in multi-horizon sensor-network forecasting (Sakhinana et al., 2024).

5. Application domains and empirical performance

The application range is broad, spanning infrastructure networks, climate, cryosphere, industrial time series, and space weather.

In power-grid and traffic forecasting, DeepGraphONet reports on the IEEE 16-machine 68-bus system and METR-LA. For the power-grid task, training on a 6-node sub-graph and testing with zero-shot transfer to a 34-node sub-graph, the reported mean L1 relative errors are 0.21% at $X(t)\in\mathbb{R}^{d\times |V|}$2 ms, 0.64% at $X(t)\in\mathbb{R}^{d\times |V|}$3 ms, 0.71% at $X(t)\in\mathbb{R}^{d\times |V|}$4 ms, and 2.57% at $X(t)\in\mathbb{R}^{d\times |V|}$5 ms; the zero-shot transfer error across sampled nodes is 0.686% (Sun et al., 2022). For traffic forecasting with $X(t)\in\mathbb{R}^{d\times |V|}$6 min, the reported means are 5.52% at $X(t)\in\mathbb{R}^{d\times |V|}$7 min, 7.60% at $X(t)\in\mathbb{R}^{d\times |V|}$8 min, and 10.79% at $X(t)\in\mathbb{R}^{d\times |V|}$9 min (Sun et al., 2022).

In ENSO forecasting, GraphINO reports all-season correlation coefficients on GODAS of 0.9747, 0.9461, 0.9170, 0.8742, 0.8226, and 0.7800 for lead months 1 through 6, with 0.6313 at 9 months, 0.5755 at 12 months, and 0.3363 at 23 months (Cachay et al., 2021). It also reports a correlation of $$\frac{d}{dt}\,\mathbf{x}(t) = f(\mathbf{x}(t); G),\quad \mathbf{x}(t_0) = \mathbf{x}_0,$$0 for forecasts issued January 2015 over the 2015/16 extreme El Niño plume predictions (Cachay et al., 2021).

In finite-element ice emulation, the Pine Island Glacier GCN reproduces ice velocity with RMSE = 55.31 m/year and Pearson correlation $$\frac{d}{dt}\,\mathbf{x}(t) = f(\mathbf{x}(t); G),\quad \mathbf{x}(t_0) = \mathbf{x}_0,$$1, and ice thickness with RMSE = 42.95 m and $$\frac{d}{dt}\,\mathbf{x}(t) = f(\mathbf{x}(t); G),\quad \mathbf{x}(t_0) = \mathbf{x}_0,$$2; the GPU-based GCN emulator is reported as approximately 33.8× faster than the CPU-based ISSM run used for comparison (Koo et al., 2024). The EGCN variant reports velocity RMSE 95.66 m/yr and thickness RMSE 27.55 m on Helheim Glacier, and velocity RMSE 55.29 m/yr and thickness RMSE 14.75 m on Pine Island Glacier, together with 260 times and 44 times faster computation time, respectively (Koo et al., 2024).

In space weather, the deterministic graph emulator produces a next step in approximately 0.47–0.52 s on 1 GPU, while the probabilistic ensemble model with ensemble size 5 takes approximately 3.2–3.45 s; the corresponding speedups relative to Vlasiator per simulated second are approximately 500× and 80× (Holmberg et al., 23 Sep 2025).

In large-scale demand forecasting, GEANN improves overall relative quantile loss on a 100K-product dataset and yields larger gains for cold-start subsets on a dataset with over 2 million products, including 0.975, 0.910, and 0.943 for P50, P90, and overall loss on newly launched items when compared to MQ-CNN (Yang et al., 2023).

6. Limitations, misconceptions, and open directions

Several limitations recur across the literature. DeepGraphONet assumes a fixed graph during forecasting and relies on a decaying memory window $$\frac{d}{dt}\,\mathbf{x}(t) = f(\mathbf{x}(t); G),\quad \mathbf{x}(t_0) = \mathbf{x}_0,$$3 that is sufficiently informative; performance can degrade if the true memory is longer, under distribution shift, or when very long-horizon forecasts must be generated without teacher-forced chunking (Sun et al., 2022). GraphINO concatenates a short input window rather than explicitly modeling temporal dynamics, uses static connectivity, and underpredicts extreme ENSO magnitudes at long leads (Cachay et al., 2021). Autoregressive simulators such as the space-weather emulator and EGCN-based ice emulator accumulate rollout error over time, even when hierarchical meshes or dynamic edge features mitigate drift (Holmberg et al., 23 Sep 2025).

Uncertainty handling is uneven. DeepGraphONet is a point-estimate model, though ensembles, MC dropout, or operator Bayesianization are identified as possible additions (Sun et al., 2022). The probabilistic space-weather emulator shows that ensemble generation can reduce RMSE of the ensemble mean at longer horizons, but the reported spread remains under target error, so the model is underdispersed (Holmberg et al., 23 Sep 2025). MKH-Net explicitly models time-varying uncertainty of multi-horizon forecasts through heteroscedastic Gaussian outputs, showing that uncertainty-aware graph forecasting can be integrated directly into the emulator formulation (Sakhinana et al., 2024).

Another misconception is that graph structure is either fixed from domain knowledge or learned entirely from data. The surveyed works include explicit graphs, learned static adjacencies, evolving layer-wise adjacencies, implicit hypergraphs, dual hypergraphs, and inductive transfer across unseen sub-graphs (Cachay et al., 2021). A plausible implication is that future progress will depend less on choosing between “fixed” and “learned” graphs than on matching graph construction, rollout strategy, and supervision scheme to the temporal scale and stability requirements of the target system.