Graph Neural Networks for Astrophysics and Cosmology

Christian Kragh Jespersen

astro-ph.IMOct 19, 2026

Graph Neural Networks for Astrophysics and Cosmology

Authors: Christian Kragh Jespersen

Organizations: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

Abstract

Physics seldom makes breakthroughs when we consider objects only in isolation. Instead we model and measure how objects sit among their neighbours, spatial, temporal, or otherwise: a galaxy in its environment, a halo in its history, a photon hit in a detector. Graphs are ideally suited for encoding exactly these relations, and Graph Neural Networks (GNNs) learn the geometric functions that respect them. This review presents the use of GNNs for astrophysics and cosmology, and argues that GNNs often suit physics and physical data better than any alternative: they fit the native geometry of the domain and the signal, enable the incorporation of the symmetries and prior knowledge we already trust and have toiled to build, and stay flexible enough to capture what we do not yet understand. GNNs' physically intuitive workings also render them more interpretable than other deep-learning architectures. We show how we can begin from general principles of the geometry of deep learning and build the graph and the graph neural network from the ground up, and how other architectures, like the convolutional network and the transformer, can be seen as special cases of one unified geometric blueprint. We survey applications across cosmology, galaxy formation, dynamics, and instrumentation; and we chart the frontier. From this survey, we find that only one in six articles outside instrumentation applies GNNs to data rather than just simulations. We also distil a set of best practices, including when not to reach for a graph. Throughout, we will prioritise physical intuition and foundational principles over mathematical machinery whenever possible.

Figures & tables

Figure 1: The three geometries of a learning problem . Each column adds one: the domain where the data live, the signal that exists upon it, and that of the model . In a well-motivated case, the domain and signal imply the model to be used. The rows fall from the abstract to two concrete cases. A galaxy sits on a pixel grid, whose translations and right-angle rotations a convolutional network inherits before a global pooling returns one rotation-invariant number, here the time since its last merger (but generalisable to most scalars). A halo catalogue forms an irregular point cloud whose only native symmetry is permutation; a linking length adds edges, and with them a bias towards local influences. We should choose the model whose symmetry respects the domain without excessive preprocessing, and is able to fully express the signal. At a minimum, it must not outlaw the true solution. At best, it matches it exactly.

Figure 2: Invariances and equivariances in a familiar physical system . For a planetary system, vector quantities such as forces and velocities are equivariant under rotations, meaning that they transform with the rotation. Scalar quantities, such as energy or eccentricities, are invariant under such transformations. Both respond to the same symmetry, but the distinction must always be kept in mind.

Figure 3: Permutation symmetry among objects without a priori ordering . A single cosmic graph — halos joined by proximity, sized by mass — carries the same physics under relabelling, with the permutation $P$ that relates the two orderings written as a condensed permutation matrix. Reorder the nodes and the graph holds, but different outputs require either invariance or equivariance: a graph-level read-out such as the cosmic matter density $\Omega_{M}$ should stay put (invariant), while the per-halo stellar masses $M_{\star}$ travel with the relabelling (equivariant). A graph neural network must honour both, so that each halo keeps its prediction regardless of its index.

Figure 4: The anatomy of a graph. We write a graph as the triplet $G=(V,E,U)$ : a set of nodes $V$ , each carrying a feature vector $\mathbf{x}_{i}$ (here a halo’s mass, position and velocity); a set of edges $E$ , each potentially carrying a feature vector $\mathbf{e}_{ij}$ on the link between two nodes (here a separation and a relative velocity); and one global vector $U$ that speaks for the whole graph (e.g. cosmological parameters, or statistics pooled across the nodes). Left , an undirected graph: the relations run both ways, as proximity, a bond, or a mutual pull, and the adjacency matrix stays symmetric, $A=A^{\top}$ . Right , a directed graph: the relation keeps a heading, as a merger, a flow, or the arrow of time, and the adjacency turns asymmetric, $A\neq A^{\top}$ . The adjacency records what links to what, with $A_{ij}=1$ for an edge and $0$ otherwise, its rows the sending index $i$ and its columns the receiving index $j$ . $A$ is useful as a complementary description of $E$ , as it splits the pure graph topology from edge features. It also hands the graph to linear algebra through the matrix description. In the right graph, a self-loop joins a node to itself, as a single entry on the diagonal, $A_{ii}=1$ (node 1). In message passing it lets a node keep a share of its own state while it folds in its neighbours’. Without it, round upon round of aggregation can sometimes wash a node’s own identity away.

Figure 5: A dark-matter halo merger tree: the assembly history forms a graph by construction, with nodes coloured by halo mass and edges that follow the flow of time. The directed edges make causality structural: information may flow from progenitor to descendant, but never back against the arrow of time. The merger tree is furthermore condensed so that only merger nodes are preserved, providing a strong inductive bias towards the importance of mergers. Adapted from Jespersen et al. (2022) .

Figure 6: The cosmic web as a proximity graph over halos. Here the graph joins halos above $10^{11}\,M_{\odot}$ within a linking length of $3$ Mpc. k-NN and Delaunay constructions offer alternatives, each with its own inductive bias (§ 3.2 ).

Figure 7: The three canonical families of GNN layers . The convolutional layer is a special case of the attentional layer, which in turn is a special case of the full message passing layer. As the degree of expressivity increases from convolutional to full message passing, the layer becomes harder and more expensive to train. We must always carefully consider this trade-off.

Figure 8: GNNs can be used across node-, edge-, and graph-level tasks , for both classification and regression (the figure annotates the classification case). This versatility makes GNNs useful across most of astrophysics and astronomy. Figure reproduced after Veličković (2023) with permission.

Figure 9: How to use edge, node, and global features in one graph-network layer : the edge updates from its endpoints, the node from its incident edges, and the global feature from the whole graph. Figure reproduced from Chuang et al. (2024) , itself closely inspired by the graph-network block of Battaglia et al. (2018) . The notation differs slightly from the one used here, but the compositions should still be clear.

Figure 10: A taxonomy of GNN computational modules , restricted to an astrophysically useful subset, with a representative reference on each leaf. Adapted in the spirit of Zhou et al. (2020) .

Table 11

Figure 11: Robustness in cosmological parameter inference with GNNs . Field-level inference of cosmological parameters survives a change of galaxy-formation model when a GNN is the backbone. Each panel shows the difference between the true and the inferred $\Omega_{m}$ for a GNN that reads only galaxy positions and line-of-sight velocities $(\mathbf{r},v_{z})$ . Trained on the Astrid simulations, the network holds its performance on simulations it never saw: SWIFT-EAGLE (top) and the cosmic-variance sets of Astrid, SIMBA, IllustrisTNG, and Magneticum (bottom). The authors attribute the robustness to carefully invariantised inputs (§ 4.6 , § 6.1.2 ). Figure 2 of de Santi et al. (2023) , reproduced from the published article at doi:10.3847/1538-4357/acd1e2 (ApJ 952, 69), © 2023 The Author(s), under the CC BY 4.0 licence.

Figure 12: The environmental galaxy–halo connection is captured by the GNN . The error (RMSE, dex) on predicted galaxy stellar mass against the linking length

L

of the environment graph. A GNN applied to a halo proximity graph with changing linking length reveals how much information each scale contains. The explainable-boosting-machine baselines, fed scalar summaries of the environment (the overdensity, DisPerSE cosmic-web features), cannot reproduce the environment connection with a similar fidelity. Note that even at

L=10

Mpc, the edge of the tested range, the GNN error still falls. Information remains to be gained from still larger scales. Reproduced from Wu et al. (2024) .

Figure 13: The solar system embedded as a graph . The Sun, the planets, Pluto, and the largest moons form the nodes, drawn from thirty years of NASA HORIZONS ephemerides sampled every thirty minutes; the messages learn the pairwise gravitational interaction. Trained on these orbits, the interaction network learns the dynamics, and symbolic regression over its messages gives us back Newton’s law of gravitation together with the masses of the bodies. Reproduced from Lemos et al. (2023) under the CC BY 4.0 licence.

Figure 14: Example event in the IceCube detector , the irregular sensor array that graph networks read natively. The size of spheres represents the amount of light observed in each digital optical module; larger spheres correspond to more light observed. The dotted black line represents the incoming neutrino, the black circle the cascade interaction point, and the solid black line the outgoing muon track. The size of the black round marker is proportional to the energy of the cascade at the vertex. The colour represents the timing of the hit, where the earliest hits are coloured in red and the latest hits in blue. Reproduced from Abbasi et al. (2022a) with kind permission of The European Physical Journal (EPJ).

Panel of Figure 15 — Figure 15: A simplified set of choices to go from data to architecture : the decision tree summary of § 6.1.1 . Regularity sends the problem to the grid-native architectures. Irregular data pass through the graph-construction choice of § 3.2 , then the layer choice splits on one central question: must the message depend on the edge or the receiver? Whatever the path, start with the sum aggregator, two to three layers, and a baseline that must be beaten.

Figure 16: Oversmoothing as thermalisation . Node features on a sample community graph, coloured by value, after zero, two, five, and twelve rounds of neighbourhood averaging: the communities blur, then vanish, as the features diffuse towards a common temperature.

Figure 17: Astrophysical GNN analyses per year, coloured by subfield , with hatching marking the analyses that include real observations. The simulation-to-reality gap, quantified from the bibliography of this review, is readily evident. The real-data fraction runs highest in instrumentation, detector reconstruction above all, while the scientific analyses remain overwhelmingly simulation-bound: the frontier this section argues the field must now cross. Generated from the works surveyed in § 5 .

Analysis	Year	Subfield	Real obs.
\svhline
Makinen et al. (2022)	2022	Cosmology & LSS	No ( ✗ )
Villanueva-Domingo & Villaescusa-Navarro (2022)	2022	Cosmology & LSS	No ( ✗ )
Adams et al. (2023)	2023	Cosmology & LSS	No ( ✗ )
de Santi et al. (2023)	2023	Cosmology & LSS	No ( ✗ )
Jung et al. (2023)	2023	Cosmology & LSS	No ( ✗ )

Table 3: The astro-GNN analyses surveyed in this review, as classified for Figure 17 . ✓ marks an analysis that includes real observations, ✗ one demonstrated on simulations alone.

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