10.4 Evolution: MEGNet, ALIGNN, M3GNet

CGCNN appeared in 2018 and demonstrated the basic point: graph neural networks reach DFT-level accuracy on bulk property prediction at orders of magnitude lower inference cost. Five years of refinements then followed, each motivated by a specific architectural shortcoming. This section traces the line through three landmark architectures — MEGNet (2019), ALIGNN (2021) and M3GNet (2022) — and is candid about what each is and is not good for.

Key Idea (Box 10.4.A)

Each post-CGCNN architecture identifies a specific shortcoming and fixes it. MEGNet adds a global state for conditioning on external variables (T, P). ALIGNN adds a line graph to inject bond angles. M3GNet adds three-body terms and trains on the full MP relaxation trajectory set, becoming the first credible universal MLIP. Each fix costs compute and parameters; choose by the physics of your target.

10.4.1 MEGNet: state attributes

CGCNN as we built it has no mechanism to condition predictions on external variables. The temperature, pressure or applied field at which a property is measured is simply absent from the model. For static ground-state properties — formation energy, band gap at zero temperature, equilibrium lattice constants — this is fine. For properties that depend on thermodynamic state — finite-temperature heat capacity, thermal conductivity at 300 K, pressure-induced phase transitions — it is a fatal omission.

Chen, Ye, Zuo, Zheng and Ong (2019) addressed this with MEGNet (MatErials Graph Network). The architecture is a CGCNN with an added global state vector \(s\) that participates in every message-passing operation. The state vector is, in the simplest case, a one-dimensional container for "the temperature at which this property was measured" or "the magnetisation of this configuration"; it can be multidimensional if several external variables are relevant.

The message-passing update in MEGNet has three stages, run in sequence at each layer.

Edge update. Each edge feature is refined using the current node states and the global state: $$ e_{uv}^{(t+1)} = \phi_e!\left( e_{uv}^{(t)},\, h_u^{(t)},\, h_v^{(t)},\, s^{(t)} \right). $$

Node update. The classical message passing, but with the global state appended: $$ h_v^{(t+1)} = \phi_h!\left( h_v^{(t)},\, \sum_{u \in \mathcal{N}(v)} e_{uv}^{(t+1)},\, s^{(t)} \right). $$

State update. The state itself evolves by aggregating over the whole graph: $$ s^{(t+1)} = \phi_s!\left( s^{(t)},\, \frac{1}{|E|}\sum_{(u,v) \in E} e_{uv}^{(t+1)},\, \frac{1}{|V|}\sum_{v \in V} h_v^{(t+1)} \right). $$ The functions \(\phi_e, \phi_h, \phi_s\) are MLPs.

The architectural lesson is general: every level — edge, node, graph — can carry its own state and be updated. CGCNN updates only nodes; MEGNet updates all three. The cost is more parameters and slower forward passes; the benefit is the ability to make state-conditioned predictions.

Derivation: how the state vector enters a single message

Spelling out the MEGNet edge update at the level of inputs and outputs makes the difference with CGCNN concrete. CGCNN's message function is \(M(h_v, h_u, e_{uv})\) — three inputs. MEGNet's edge update is $$ e_{uv}^{(t+1)} = \phi_e!\left([\,h_u^{(t)};\, h_v^{(t)};\, e_{uv}^{(t)};\, s^{(t)}\,]\right), $$ where the concatenated input has dimension \(2d + d_e + d_s\) rather than \(2d + d_e\) (CGCNN). The additional \(d_s\) components carry the global state into every edge computation. If \(s = (T)\) is just temperature, \(d_s = 1\) and the model sees the same temperature on every edge — equivalent to passing \(T\) as an extra input feature to a CGCNN that has been retrained per temperature.

The trick that pays off is that the same trained MEGNet works for all temperatures: training data at \(T = 100, 300, 500\) K is pooled, and at test time we set \(s = T_\text{test}\) and predict. CGCNN would require three separate models or temperature as a node feature (with the same effective consequence but more awkward bookkeeping).

Numerical example: temperature-dependent heat capacity

Suppose we train MEGNet on Materials Project heat-capacity data at 100, 200, 300 K and want to predict at 250 K. After training, set \(s^{(0)} = 250\) (after any normalisation the model used during training) and run the forward pass. Because every layer's update uses \(s\), the prediction smoothly interpolates between the network's behaviour at the training temperatures. With CGCNN we would either need separate networks or to retrain with \(T\) as a seventh node feature — equivalent but less elegant.

MEGNet's headline result was a single network trained on \(\sim 64\,000\) Materials Project crystals reaching \(0.028\) eV/atom MAE on formation energies, comfortably beating CGCNN's \(\sim 0.04\) eV/atom on the same benchmark. More striking was that MEGNet matched or beat specialised SchNet variants on QM9 molecules using the same architecture with no modifications, suggesting that the inductive bias was right.

When to use MEGNet rather than CGCNN: when your target depends on a continuous external variable (temperature, pressure, doping), or when you want a single model to predict several properties at once and you need a place to inject the property-identity tag. For one-property, zero-temperature regression, CGCNN gets you most of the way there.

10.4.2 ALIGNN: bond angles via the line graph

A persistent weakness of all the architectures so far — CGCNN, SchNet, MEGNet — is that they see only distances between atoms. A pair of crystals with identical sets of interatomic distances but different bond angles would be assigned identical embeddings. In practice such ambiguous pairs are rare, but the missing angular information leaves accuracy on the table: bond angles encode hybridisation and local coordination geometry, both highly informative.

The Atomistic Line Graph Neural Network (ALIGNN) of Choudhary and DeCost (2021) fixes this with an elegant construction. They observe that a graph's edges can themselves be promoted to nodes of a new graph, the line graph. In the line graph, each node corresponds to a bond in the original; two line-graph nodes are connected by a line-graph edge if the two original bonds share an atom. The line-graph edge then naturally carries the bond angle between the two bonds.

Formally, given \(G = (V, E)\), the line graph \(L(G) = (E, E')\) has node set \(E\) and edge set \(E' = \{(e_1, e_2) : e_1, e_2 \in E, e_1 \cap e_2 \neq \emptyset\}\). Each line-graph edge \((e_1, e_2)\) — where \(e_1 = (u, v)\) and \(e_2 = (v, w)\) share atom \(v\) — carries the angle \(\theta = \angle uvw\) expanded in a Gaussian basis.

Worked construction: line graph of a triangle

Take \(G\) as a triangle with nodes \(\{1, 2, 3\}\) and edges \(e_{12}, e_{23}, e_{13}\) (treated as undirected). The line graph \(L(G)\) has three nodes (one per original edge). Edges between them: \(e_{12}\) and \(e_{23}\) share atom 2; \(e_{23}\) and \(e_{13}\) share atom 3; \(e_{12}\) and \(e_{13}\) share atom 1. So \(L(G)\) is itself a triangle — every pair of original edges shares an atom.

Each \(L(G)\)-edge carries a bond angle. Suppose the original triangle has angles \(60°, 60°, 60°\) at atoms 1, 2, 3 respectively. Then the line-graph edges between (12,13), (12,23), (13,23) carry those three angles, expanded in a basis.

ALIGNN then runs MPNN on \(G\) to update edge features, then MPNN on \(L(G)\) (with edge features as nodes) to incorporate angle information, alternating. After several alternations, the central atom's embedding has absorbed information about both its bond lengths and its bond angles — the missing chemistry of coordination geometry.

Why angles distinguish polymorphs

Anatase and rutile TiO\(_2\) share the same composition and similar Ti–O bond lengths (around 1.94 Å in both). They differ in the bond angles: rutile's TiO\(_6\) octahedra are nearly regular (angles near 90°), while anatase's are distorted (angles around 78° and 102°). A distance-only GNN like CGCNN cannot distinguish them on geometry alone — only on the slightly different distributions of bond lengths and the topology of the edge graph, which is a much weaker signal. ALIGNN, by including angles explicitly, makes the distinction sharp. This is why ALIGNN's advantage over CGCNN is largest on polymorph-rich datasets.

ALIGNN then runs message passing alternately on the original crystal graph and on its line graph: each block updates the bond representations using both adjacent atoms and adjacent angles, and the atom representations using updated bonds. After several alternating blocks, the atom embeddings have absorbed angular information through the intermediate bond representations.

The accuracy gain is real and consistent. On the Materials Project formation-energy benchmark, ALIGNN reaches \(0.022\) eV/atom MAE versus MEGNet's \(0.028\) and CGCNN's \(0.039\). On bulk modulus, shear modulus and many other elastic properties the gap is larger — angles matter more for mechanical stiffness than for energetics. As of writing, ALIGNN remains the strongest published architecture on the full Matbench suite of property-regression tasks.

The cost is computational. The line graph has up to \(|E| \times \bar{k}\) edges (where \(\bar{k}\) is the average node degree), which can be ten times larger than the original edge count. ALIGNN training takes roughly four times longer than CGCNN training for the same number of epochs, and the memory footprint is higher. For high-throughput screening this matters; for one-off model training it does not.

When to use ALIGNN: when angular information is plausibly important (elastic properties, polymorph energetics, anything involving hybridisation changes) and you have the compute. For applications where inference speed matters — embedding millions of candidates — a faster model like CGCNN with the trade-off explicitly accepted may be the right call.

10.4.3 M3GNet: three-body terms and a universal MLIP

Chen and Ong's 2022 M3GNet paper makes a different bet. The authors set out not to write the best property-regression GNN — ALIGNN already existed — but to write the first credible universal machine-learning interatomic potential, capable of predicting energies, forces and stresses for any element in the periodic table.

To do this they took the line-graph idea, refined it as the inclusion of explicit three-body terms in the message-passing scheme, and trained on the full Materials Project relaxation trajectory set — every intermediate geometry produced during the structural optimisation of every entry in the database.

What 'explicit three-body terms' actually means

A standard two-body MPNN computes messages that depend on \((h_v, h_u, r_{uv})\) — pairwise. A three-body term depends on three atoms simultaneously: \((h_v, h_u, h_w, r_{uv}, r_{vw}, \theta_{uvw})\). The simplest way to inject this into an MPNN is to compute, for each pair of edges \((uv, vw)\) sharing atom \(v\), a three-body message $$ m_{uw|v}^{(3b)} = \phi_3!\left( h_v, h_u, h_w, e_{uv}, e_{vw}, e_{uvw}^{\text{angle}} \right), $$ and aggregate over all such triples into the central atom. Formally this is a tensor-field-network-style construction: the three-body message transforms as a scalar under rotation if \(\phi_3\) uses only invariants (\(r_{uv}, r_{vw}, \theta_{uvw}\)), or as a higher-rank tensor if \(\phi_3\) involves spherical harmonics of the unit vectors \(\hat{\mathbf{r}}_{uv}\) and \(\hat{\mathbf{r}}_{vw}\) (Chapter 9, §9.2). The training set is about 187 000

relaxation trajectories with \(\sim 1.6\) million single-point calculations. The model is trained jointly on energy, force and stress losses (Chapter 9, §9.4).

The result is a single model with about \(\sim 0.7\) million parameters that runs on a laptop at hundreds of structures per second and reaches formation-energy MAE of \(0.035\) eV/atom (below the DFT error itself for many materials), force MAE of about \(70\) meV/Å, and supports geometry relaxations of arbitrary crystals out of the box.

M3GNet's significance is conceptual rather than architectural. It is the first piece of evidence that one can have a foundation MLIP — a single network trained once, fine-tuned for specific systems, and reused as a drop-in replacement for DFT in geometry optimisations and short MD runs. Chapter 12 will revisit this when we discuss CHGNet, MACE-MP-0 and the broader foundation-model story.

When to use M3GNet (or its successors): whenever you need forces, not just energies, and you are willing to accept errors of around 100 meV/Å in those forces. M3GNet is the go-to for high-throughput relaxation — take a database of CIF files, run M3GNet relaxation on each, and you have approximately-DFT-converged geometries in seconds per structure.

When not to use it: when you need DFT-level accuracy on a specific material class. M3GNet's universal training distributes its capacity across the periodic table, and a specialised model fine-tuned on, say, iron-based superconductors will beat M3GNet on iron-based superconductors. The relationship is exactly the foundation-model-and- fine-tune pattern of modern machine learning.

10.4.4 An honest comparison

Property	CGCNN	MEGNet	ALIGNN	M3GNet
Formation energy (eV/atom)	0.039	0.028	0.022	0.035
Band gap (eV)	0.388	0.330	0.218	0.330
Bulk modulus (\(\log_{10}\) GPa)	0.071	0.060	0.051	0.068
State-conditioned predictions	no	yes	no	no
Forces / stresses available	no	no	no	yes
Inference speed	fast	fast	slow	medium
Training speed	fast	fast	slow	medium
Code complexity	low	medium	high	high
Pre-trained universal weights	no	partial	partial	yes

Numbers above are from the published papers on the Materials Project test split as of late 2024; details and exact protocols vary, and the relative ordering is more meaningful than the absolute values.

Matbench formation-energy MAE — exact comparison

The Matbench matbench_mp_e_form task (132 000 stable inorganic crystals, 5-fold cross-validation, fixed splits, formation energy per atom) gives a cleaner comparison than the Materials Project test split. Approximate published MAE values (eV/atom) at the time of writing:

Architecture	MAE	Year
CGCNN	0.039	2018
MEGNet	0.028	2019
SchNet	0.034	2018
ALIGNN	0.022	2021
M3GNet	0.025	2022
MACE-MP-0	0.020	2023

The trend is clear: each generation has dropped MAE by 5–25%, and we are now within a factor of two of the DFT noise floor itself (different functionals disagree by 30–50 meV/atom on the same structure). Further architectural gains are likely small; the leverage has shifted to data — larger, cleaner, more diverse training sets — and to fine-tuning foundation models on specialised subsets, the subject of Chapter 12.

The reader who has followed §10.3 to a working CGCNN now has the right foundation to read the MEGNet, ALIGNN and M3GNet papers. Each is a specific instantiation of the abstract MPNN framework of §10.2; each trades complexity for accuracy or generality. The right choice depends on the question being asked.

10.4.5 What the field is currently doing

Beyond the four architectures above, three threads dominate recent work. The first is equivariance: NequIP and MACE (covered in Chapter 9) extend M3GNet's universal-MLIP ambition with explicit \(\mathrm{SO}(3)\)-equivariant features, and the resulting models (MACE-MP-0, MACE-OFF) are competitive with M3GNet at lower parameter counts. The second is scale: the Open Catalyst Project's OC20 and OC22 datasets, with twenty million single-point calculations, have become a benchmark for how large materials GNNs can be trained. GemNet-OC and equivariant successors push these scales further. The third is foundation models for graphs: pre-trained universal backbones (CHGNet, MACE-MP-0, ORB, MatterSim) that fine-tune on small target datasets. Chapter 12 takes up this story in detail.

The mood of the field has shifted accordingly. The question is no longer "what architecture should I train from scratch on my \(10^3\) crystals?" — it is "which pre-trained foundation model should I fine-tune?" CGCNN, MEGNet and ALIGNN remain valuable both as pedagogical examples and as efficient task-specific models when you do train from scratch. But the production answer in 2026 is increasingly to start from a foundation model and adapt it. Section 10.5 considers what that adaptation looks like in practice on the Materials Project, and what subtleties — particularly around train/test splitting — make the difference between honest accuracy estimates and self-flattering ones.

10.4.5a A decision tree for choosing an architecture

The reader confronting a new property-regression task will rationally ask: which of these four should I train? A pragmatic decision tree.

Do I need forces or stresses? If yes (geometry optimisation, molecular dynamics, phonons), use M3GNet or MACE. If no, continue.

Do I have a state variable to condition on (temperature, doping fraction)? If yes, use MEGNet. If no, continue.

Are angles likely to matter? (polymorph energetics, elastic properties, anything involving hybridisation changes)? If yes, use ALIGNN. If no, use CGCNN.

Do I have \(< 10^4\) training labels? In all cases, prefer fine-tuning a foundation model (CHGNet, MACE-MP-0; Chapter 12) over training from scratch. The pre-trained representations carry most of the chemistry your training labels would otherwise have to teach.

This decision tree is not exhaustive — in practice you should also consider training time, hardware constraints, and the availability of reference implementations — but it covers 80% of real choices. For the remaining 20%, ablation studies on your specific data are the only honest answer.

10.4.5b What changes if I want forces?

Returning briefly to the forces question. A CGCNN trained on formation energies produces \(\hat{E}(\text{structure})\). To get forces one would naively autodifferentiate: \(\hat{\mathbf{F}}_i = -\nabla_{\mathbf{r}_i} \hat{E}\). The technical problem is that CGCNN's edge features depend on distances \(r_{uv}\), not on coordinates directly, and the graph itself (neighbour assignments) depends on coordinates via the cutoff. The latter dependence is discontinuous: as an atom crosses the cutoff sphere of a neighbour, the graph topology changes abruptly, and the forces blow up.

The fix used by M3GNet and successors is to smoothly attenuate messages near the cutoff using a polynomial envelope \(f_{\text{env}}(r) = (1 - r/r_c)^p\) for \(r < r_c\), zero beyond. Messages then vanish smoothly as atoms approach the cutoff, and the gradient of \(\hat E\) with respect to coordinates is well-defined and continuous. This is one of several technical reasons that a force-predicting GNN is meaningfully harder to build than an energy-predicting one. Chapter 9 develops the details.

Cross-reference

Chapter 9's MLIP discussion covers the smoothness-of-cutoff question, equivariance, and force-loss design. The MPNN architectures here are the structural basis for those MLIPs; the training differences are covered there.

10.4.6 A note on benchmarking

Every number in this section depends on the test split, the data preprocessing and the hyperparameter budget. Comparisons in the literature are not always apples-to-apples. The Matbench benchmark (Dunn et al., 2020) is the closest the field has to a standardised playing field for property-regression GNNs; it specifies fixed train/test splits and evaluation protocols across thirteen tasks and publishes a public leaderboard. If you are choosing between architectures for a real project, look at Matbench rankings filtered to the property class you care about — not the headline number from a single paper.

We will return to the benchmarking question in §10.5, where the same pitfall reappears in a different guise: random splits of a polymorph- rich database systematically overstate model accuracy. The fix is the structurally disjoint split, and the difference can be a factor of two or three in reported MAE.

A reproducible benchmarking workflow

To compare two architectures honestly on your own data:

1. Use exactly the same train/val/test split for both.
2. Use the same featurisation (cutoff, basis size, etc.).
3. Allocate the same wall-clock training budget — do not let a slow architecture run for longer at the expense of a faster one.
4. Report a mean and standard deviation over \(\geq 3\) random seeds. A single-seed comparison routinely produces 10% spread between runs of the same architecture.
5. Specify the exact MP database version (or commit hash of your data file). The same query can return different structures weeks apart.

This is more bookkeeping than most papers do. Doing it nevertheless will reveal that on small datasets the four architectures are often within statistical noise of each other, and on large datasets the gaps are real but smaller than the advertised headline numbers.

Section summary

• MEGNet adds a global state vector that participates in every message-passing update; the right choice for state-conditioned prediction (T, P, doping).
• ALIGNN promotes bonds to nodes of a line graph carrying bond angles, gaining accuracy on polymorphs and elastic properties at \(\sim 4\times\) training cost.
• M3GNet adds three-body terms and trains on relaxation trajectories, yielding the first credible universal MLIP for forces and stresses.
• Choose the architecture by the physics of the target — angles, forces, state-dependence — not by headline benchmark numbers.