Materials Genomics
Unit 7: Graph-Based Crystal Representations
FAU Erlangen-Nürnberg
§1 · Where We Stand
01. Unit goal
- Move from fixed descriptors (U6 last week — Magpie / RDF / ACSF / SOAP) to learned representations
- Treat a crystal as a graph: atoms = nodes, bonds = edges, periodicity = edge wrap-around
- Build the math toolkit (PBC, RBF, invariance / equivariance, message passing) that U9, U10 will rely on
02. Learning objectives
By the end of this unit, students can:
- Define a periodic crystal graph \(G=(V,E)\) with lattice-vector-aware edges
- Implement PBC via the minimum-image convention and explain why naive cutoffs fail
- Expand interatomic distances in a Gaussian RBF basis and reason about smooth cutoffs
- Distinguish invariance (energy) from equivariance (forces, dipoles)
- Write the generic message-passing update and instantiate it for CGCNN, MEGNet, SchNet
- Choose readout consistent with extensive vs intensive targets and diagnose over-smoothing
03. Recap: prerequisite map
- U2: Schrödinger / DFT — generates the labels (energies, forces)
- U3: Crystal lattices, space groups, primitive cells
- U6 (last week): Hand-crafted atom-centred descriptors (Magpie, RDF, ACSF, SOAP, ACE) — the classical analogue of today’s learned graphs
- MFML W4-5: Feedforward NNs and backprop
Today’s step
Hand-crafted features \(\Rightarrow\) learned features by neighbour-aware message passing.
04. Why this matters: failure modes of fixed descriptors
- Descriptors require a fixed-size vector \(\Rightarrow\) varying \(N\), varying stoichiometry break the input shape
- Choice of cutoff, RBF grid, and ordering is manual and brittle
- Permutation invariance enforced by hand (sort, average) \(\to\) information loss
- Graphs absorb all three: variable size, permutation invariance, and learnable edge features
- Bonus: graphs are data-efficient — weight sharing across atoms means few parameters per node
05. Reading map
- Core theory: Sandfeld Ch. 2.2 (structure encoding)
- Architectures: Neuer Ch. 4.5.1–4.5.4 (GNNs for engineering)
- Deep learning reference: Murphy Ch. 35 (graph neural networks)
- Equivariance: Bronstein et al., Geometric Deep Learning (2021)
§2 · Crystals as Graphs
06. Crystals as periodic graphs
Crystal structure \(\to\) graph \(G = (V, E)\) over the primitive unit cell plus periodic images.
- Nodes \(V\): atoms in the primitive cell, \(|V| = N\)
- Edges \(E\): pairs \((i, j, \vec{n})\) with \(i, j \in V\), image \(\vec{n}\in\mathbb{Z}^3\), within cutoff \(r_c\)
- Node features \(h_i\): atomic number \(Z_i\), valence, electronegativity, group, period, oxidation state
- Edge features \(e_{ij}\): displacement \(\vec{r}_{ij}\), distance \(d_{ij}\), optionally bond type / order
With lattice matrix \(\mathbf{T}=[\vec{a}_1,\vec{a}_2,\vec{a}_3]\in\mathbb{R}^{3\times 3}\), the lattice-aware displacement to image \(\vec{n}\) is
\[\vec{r}_{ij}(\vec{n}) \;=\; \vec{r}_j + \mathbf{T}\,\vec{n} - \vec{r}_i, \qquad d_{ij}(\vec{n}) = \|\vec{r}_{ij}(\vec{n})\|\]
Edge created iff \(d_{ij}(\vec{n}) \le r_c\) — a single ordered pair \((i,j)\) may yield several edges (one per image inside the cutoff sphere).
07. Periodic boundary conditions (PBC)
A finite-cell graph must reproduce the topology of the infinite lattice.
Minimum-image convention: replace the raw displacement by its closest periodic image,
\[\vec{r}_{ij} \;\longleftarrow\; \vec{r}_{ij} \;-\; \mathbf{T}\,\mathrm{round}\!\big(\mathbf{T}^{-1}\,\vec{r}_{ij}\big)\]
where \(\mathrm{round}\) acts component-wise on the fractional displacement.
Why a naive cutoff fails on small cells:
- Primitive NaCl cell \(a \approx 5.64\) Å, two atoms (Na at origin, Cl at \(\tfrac{a}{2}(\hat{x}+\hat{y}+\hat{z})\))
- Cutoff \(r_c = 4\) Å misses several of Na’s six nearest Cl neighbours unless periodic images are searched
- Solution: tile the cell into a \(3\times 3\times 3\) supercell during neighbour search, or use the minimum-image rule plus image-index bookkeeping
Forgetting PBC \(\Rightarrow\) disconnected graph, missing nearest neighbours, wrong coordination — the most common silent bug in custom pipelines.
08. Graph construction workflow
CIF / POSCAR
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parse lattice T and fractional coords {f_i}
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Cartesian coords r_i = T · f_i
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neighbour search: for each i, j, image n ∈ Z^3
compute d_ij(n); keep iff d_ij(n) ≤ r_c
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edge tensor [src, dst, image, d_ij, r_ij]
node tensor [Z_i, group, period, ...]
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batched DGL / PyG graph → GNN- Backends:
pymatgen.StructureGraph,ase.neighborlist,jarvis.core.graphs,torch_geometric.transforms.RadialGraph
09. Neighbour-cutoff strategies
Fixed cutoff \(r_c\)
- All \(j\) with \(d_{ij} \le r_c\)
- Pros: physically meaningful (bonding shell)
- Cons: variable degree; sensitive to thermal expansion / pressure
\(k\)-nearest neighbours
- Keep the \(k\) closest atoms regardless of distance
- Pros: constant degree, easy batching
- Cons: in sparse regions, “neighbours” can be unphysically far
Hybrid: \(k\)-NN with maximum cutoff (max_neighbors=12, r_c=8 Å) — the practical default in CGCNN, MEGNet, ALIGNN.
10. Pseudocode: lattice-aware neighbour list
def build_edges(frac_coords, T, r_c):
cart = frac_coords @ T # (N, 3)
# search shells of periodic images that intersect r_c-sphere
n_max = ceil(r_c / min_lattice_spacing(T))
images = product(range(-n_max, n_max+1), repeat=3)
edges = []
for n in images:
shift = np.array(n) @ T # (3,)
for i in range(N):
for j in range(N):
r_ij = cart[j] + shift - cart[i]
d = np.linalg.norm(r_ij)
if 1e-8 < d <= r_c:
edges.append((i, j, n, d, r_ij))
return edges- Cost \(\mathcal{O}(N^2 \cdot |\text{images}|)\); replace inner loops by KD-tree on the supercell for large \(N\).
§3 · Encoding Geometry
11. Distance encoding: Gaussian RBF
Raw \(d_{ij}\) is a poor input to an MLP — it is a single scalar with strong non-linearity in the energy.
Expand into a Gaussian RBF basis of \(K\) centres \(\{\mu_k\}\):
\[e_{ij,k} \;=\; \exp\!\Big(-\,\frac{(d_{ij} - \mu_k)^2}{2\sigma^2}\Big), \qquad k = 1,\ldots,K\]
Typical grid (CGCNN / SchNet):
- \(\mu_k\) from \(0\) to \(5\) Å in \(\Delta\mu = 0.1\) Å steps \(\Rightarrow K = 50\)
- Width \(\sigma = 0.2\) Å (matches \(\Delta\mu\))
- Yields a smooth, differentiable distance fingerprint for each edge
Why a smooth basis:
- Gradient \(\partial e_{ij,k}/\partial \vec{r}_j\) is well-defined \(\Rightarrow\) forces \(\vec{F}_i = -\nabla_i E\) via autograd
- Soft cutoff envelope \(f_c(d_{ij}) = \tfrac{1}{2}[\cos(\pi d/r_c) + 1]\) avoids energy jumps as atoms cross \(r_c\)
12. Edge engineering beyond pair distances
- Bond order / chemical bond type (single, double, ionic) from valence rules
- Triplet / angular features \(\theta_{ijk}\): needed for force fields, central in ALIGNN, GemNet
- Dihedrals \(\phi_{ijkl}\): relevant for organic / molecular crystals
- Coordination number \(Z_i^{\rm coord} = |\mathcal{N}(i)|\) as a node feature
- Composition-derived priors: electronegativity, ionization energy, atomic radius — initialise \(h_i^{(0)}\) instead of one-hotting \(Z\)
Higher-order geometry \(\to\) stricter inductive bias, but more expensive graphs and longer training.
§4 · Symmetries: Invariance and Equivariance
13. Symmetries that physics demands
A property predictor on a crystal must respect three groups acting on positions:
- Translation \(T_{\vec{t}}\): \(\{\vec{r}_i\} \to \{\vec{r}_i + \vec{t}\}\)
- Rotation / reflection \(R \in O(3)\): \(\vec{r}_i \to R\,\vec{r}_i\)
- Permutation \(\pi \in S_N\): relabelling of atoms
The right requirement depends on the target tensor type.
14. Invariance vs equivariance
Invariance (scalars):
\[f(\{R\vec{r}_i + \vec{t}\}) \;=\; f(\{\vec{r}_i\})\]
- Energy \(E\), bandgap \(E_g\), formation energy \(\Delta H_f\)
- Achieved by using only distances and angles as edge features
Equivariance (vectors / tensors):
\[\vec{F}\!\left(\{R\vec{r}_i + \vec{t}\}\right) \;=\; R\,\vec{F}(\{\vec{r}_i\})\]
- Forces \(\vec{F}_i\), dipole \(\vec{\mu}\), polarisation, elastic tensor \(C_{ijkl}\)
- Requires vector-valued node features that transform with \(R\)
Permutation invariance: \(f(\pi\cdot V) = f(V)\) for any \(\pi \in S_N\) — handled automatically by sum/mean readouts over nodes.
15. Why equivariance is non-negotiable for forces
Forces are gradients of an invariant scalar:
\[\vec{F}_i = -\nabla_{\vec{r}_i} E\]
If \(E\) is rotation-invariant and differentiable, then automatically
\[\vec{F}_i(R\,\{\vec{r}\}) \;=\; -\nabla_{R\vec{r}_i} E(R\,\{\vec{r}\}) \;=\; R\,\big(-\nabla_{\vec{r}_i} E(\{\vec{r}\})\big) \;=\; R\,\vec{F}_i(\{\vec{r}\})\]
- Distance-only GNNs (CGCNN, SchNet, MEGNet) predict invariant scalars and obtain forces by autograd \(\Rightarrow\) correct equivariance, but limited capacity
- Tensor-valued GNNs (NequIP, MACE, Allegro) carry \(O(3)\)-equivariant features at each node — they can predict vectors and higher-rank tensors directly
- Without equivariance, learnt forces are not closed under rotation \(\Rightarrow\) MD trajectories drift
§5 · Message Passing
16. The message-passing template
A GNN layer updates each node from its neighbourhood. Generic form:
\[\boxed{\;h_i^{(\ell+1)} \;=\; U^{(\ell)}\!\Big(h_i^{(\ell)},\;\;\mathrm{AGG}_{j\in\mathcal{N}(i)}\, M^{(\ell)}\!\big(h_i^{(\ell)},\,h_j^{(\ell)},\,e_{ij}\big)\Big)\;}\]
- \(M^{(\ell)}\): message function (MLP on neighbour and edge features)
- \(\mathrm{AGG}\): permutation-invariant aggregator (sum, mean, max, attention)
- \(U^{(\ell)}\): update function (gated MLP, GRU, residual block)
After \(L\) layers, node \(i\) has aggregated information from atoms within graph distance \(L\) — its receptive field.
17. Concrete instantiation: GCN-style update
Take the simplest crystal-graph variant:
\[h_i^{(\ell+1)} \;=\; \sigma\!\Big(W^{(\ell)}\,\big(h_i^{(\ell)} + \tfrac{1}{|\mathcal{N}(i)|}\sum_{j\in\mathcal{N}(i)} \alpha_{ij}\,h_j^{(\ell)}\big) + b^{(\ell)}\Big)\]
- \(\alpha_{ij}\): edge weight, e.g. \(\alpha_{ij} = \mathrm{MLP}(e_{ij})\) or \(\alpha_{ij} = 1/\sqrt{d_id_j}\)
- \(\sigma\): ReLU / SiLU
- Sum aggregation \(\Rightarrow\) permutation invariance for free
18. Walk-through: 3-atom toy graph
A water-like fragment \(V = \{O, H_1, H_2\}\), edges \(\{(O,H_1),(O,H_2),(H_1,H_2)\}\), scalar features \(h_i^{(0)} = Z_i\).
Layer 1 (sum aggregation, \(\alpha_{ij}=1\), \(W=1\), \(\sigma=\mathrm{id}\)):
\[h_O^{(1)} = h_O^{(0)} + h_{H_1}^{(0)} + h_{H_2}^{(0)} = 8 + 1 + 1 = 10\]
\[h_{H_1}^{(1)} = 1 + 8 + 1 = 10, \qquad h_{H_2}^{(1)} = 1 + 8 + 1 = 10\]
Layer 2: now every node sees the second-shell sum,
\[h_O^{(2)} = 10 + 10 + 10 = 30\]
— the receptive field doubles per layer. With realistic \(W\) and \(\sigma\), the same mechanism builds chemically-meaningful local descriptors.
§6 · Crystal GNN Architectures
19. CGCNN — crystal graph convolutional NN
Xie and Grossman (2018) introduce the first widely-used crystal-graph model.
- Nodes initialised with one-hot atomic-property vectors
- Edge features: Gaussian-expanded distances \(e_{ij}\)
- Gated convolution combining neighbour and edge features:
\[h_i^{(\ell+1)} \;=\; h_i^{(\ell)} + \sum_{j\in\mathcal{N}(i)} \sigma\!\big(W_z\,z_{ij}^{(\ell)} + b_z\big) \;\odot\; g\!\big(W_s\,z_{ij}^{(\ell)} + b_s\big)\]
with \(z_{ij}^{(\ell)} = h_i^{(\ell)} \Vert h_j^{(\ell)} \Vert e_{ij}\), sigmoid gate \(\sigma\) and softplus content \(g\).
- Use case: formation energy, bandgap, bulk modulus on the Materials Project
20. MEGNet — global state vector
Chen et al. (2019) add a global node \(u\) that interacts with every atom and edge.
- Edges, nodes, and the global vector are updated each layer:
\[u^{(\ell+1)} \;=\; \phi_u\!\Big(u^{(\ell)},\;\tfrac{1}{N}\!\sum_i h_i^{(\ell+1)},\;\tfrac{1}{|E|}\!\sum_{ij} e_{ij}^{(\ell+1)}\Big)\]
- \(u\) encodes state variables: temperature, pressure, applied field
- Same architecture handles molecules and crystals with a single global flag
- Use case: state-aware property prediction, multi-task learning
21. SchNet — continuous-filter convolutions
Schütt et al. (2018) replaces discrete edge weights by a continuous filter \(W: \mathbb{R}^+ \to \mathbb{R}^F\) evaluated on \(d_{ij}\).
\[h_i^{(\ell+1)} \;=\; h_i^{(\ell)} + \sum_{j\in\mathcal{N}(i)} h_j^{(\ell)} \,\odot\, W^{(\ell)}\!\big(d_{ij}\big)\]
- \(W^{(\ell)}\) is itself an MLP applied to the RBF expansion of \(d_{ij}\)
- Smooth in \(d_{ij}\) \(\Rightarrow\) analytically-differentiable energy \(\Rightarrow\) forces by autograd
- Use case: potential energy surfaces, MD with ML interatomic potentials
22. Equivariant GNNs in one slide
Distance-only models cannot represent vectorial features in their hidden layers. Equivariant GNNs carry \(O(3)\)-equivariant tensors:
- Tensor Field Networks / NequIP: hidden features are spherical-tensor-valued; messages use Clebsch–Gordan tensor products
- MACE: many-body equivariant messages via Atomic Cluster Expansion
- Allegro: strictly-local equivariant messages, no global pooling
- Cost: more expensive per layer, but far more data-efficient for forces, polarisation, elastic tensors
Rule of thumb: predicting scalars only \(\to\) CGCNN / SchNet is enough; predicting vectors / tensors \(\to\) go equivariant.
§7 · Readout, Depth, Practicalities
23. Readout / pooling
After \(L\) message-passing layers, aggregate node features into a graph-level vector \(h_G\):
\[\text{sum:}\quad h_G = \sum_{i\in V} h_i^{(L)} \qquad \text{mean:}\quad h_G = \tfrac{1}{N}\sum_{i\in V} h_i^{(L)}\]
\[\text{attention:}\quad h_G = \sum_{i\in V} \alpha_i\,h_i^{(L)}, \qquad \alpha_i = \frac{\exp(w^\top h_i^{(L)})}{\sum_j \exp(w^\top h_j^{(L)})}\]
- Extensive targets (total energy, formation energy per cell): sum preserves \(E \propto N\)
- Intensive targets (bandgap, bulk modulus, density): mean is correct; sum would scale wrongly with cell size
- Attention when contributions are sparse (defects, active sites)
The mismatched-readout pitfall: training “energy / atom” with sum readout silently learns to predict cell size.
24. Graph depth and over-smoothing
Each layer mixes neighbour features \(\Rightarrow\) after many layers, all node embeddings converge.
Empirically, \(L \ge 4\) on dense crystal graphs collapses node features:
\[\lim_{\ell \to \infty} h_i^{(\ell)} = h_j^{(\ell)} \quad \forall i, j \in V\]
— the GNN forgets which atom is which.
Mitigations:
- Residual / skip connections: \(h_i^{(\ell+1)} = h_i^{(\ell)} + \mathrm{MP}(h_i^{(\ell)})\)
- Jumping knowledge (Xu et al. 2018): readout concatenates \(h_i^{(1)}, \ldots, h_i^{(L)}\)
- DenseNet-style dense skip: every layer reads every previous layer
- Practical recipe: 3–4 message-passing layers + residual + Gaussian RBF edges — strong default for materials
25. Variable-size cells and batching
- Crystal graphs vary in \(N\) and \(|E|\) — cannot stack into a single tensor
- Disjoint-batch trick: concatenate all graphs into one large graph with block-diagonal adjacency; pooling uses a
batch_indexvector - Implemented by
torch_geometric.data.Batchanddgl.batch - Weight matrices are shared across all atoms \(\Rightarrow\) same model handles primitive cells and supercells
26. Computational scaling
- Neighbour search: \(\mathcal{O}(N \log N)\) with KD-tree, \(\mathcal{O}(N^2)\) naive
- One message-passing layer: \(\mathcal{O}(|E|)\)
- For 3D crystals at fixed \(r_c\), \(|E| \approx \bar{k}\,N\) with average degree \(\bar{k}\)
- Total cost \(\mathcal{O}(L \cdot \bar{k} \cdot N)\) — linear in cell size, vastly faster than DFT \(\mathcal{O}(N^3)\)
Multi-modal extensions (atom + microstructure + literature graphs) exist but are niche — covered briefly in U14.
§8 · Failure Modes and Diagnostics
27. Failure mode: shortcut learning
- If unit-cell volume strongly correlates with energy in the dataset, a GNN can learn to predict \(E\) from \(N\) alone, ignoring chemistry
- Diagnostic: randomise atomic numbers while keeping geometry; if \(R^2\) stays high, the model is cheating
- Diagnostic: fix chemistry, perturb cell volume; honest models follow \(E(V)\) curves
Always pair a graph baseline with a composition-only MLP to expose this.
28. Failure mode: cutoff and reproducibility artefacts
- A hard cutoff \(r_c\) creates discontinuities: as \(d_{ij}\) crosses \(r_c\), edge appears/disappears \(\Rightarrow\) jumps in \(E\), divergent forces
- Smooth envelope \(f_c(d_{ij}) = \tfrac{1}{2}[\cos(\pi d_{ij}/r_c) + 1]\) for \(d_{ij} \le r_c\), \(0\) otherwise — multiply edge messages by \(f_c\)
- Small changes in \(r_c\) (e.g. \(4.0 \to 4.5\) Å) can flip graph topology — always log \(r_c\), RBF grid, neighbour-search algorithm
Reproducibility checklist: (i) cutoff, (ii) max neighbours, (iii) RBF \(\{\mu_k, \sigma\}\), (iv) PBC convention, (v) random seed.
29. Transferability and OOD behaviour
Transfer across chemistries
- Oxides \(\to\) nitrides: feasible if elemental-feature initialisation is rich
- Pure \(Z\)-embedding \(\to\) poor; physical features (electronegativity, radius) \(\to\) better
Out-of-distribution structures
- Perovskite-trained \(\to\) spinel: GNNs degrade gracefully but predictions are biased
- Use ensemble disagreement as a novelty flag
- Detailed UQ machinery: U13
30. Baseline comparison protocol
When reporting a new GNN result, always compare against:
- Mean-target baseline (the trivial predictor)
- Composition-only MLP (no geometry) — exposes shortcut learning
- Classical descriptor baseline (PRDF / SOAP + ridge or RF)
- Strong public GNN (CGCNN or MEGNet trained on the same split)
Any new architecture must beat all four on both MAE and Spearman rank correlation.
31. Evaluation metrics for screening
- MAE / RMSE: regression accuracy
- Spearman \(\rho\) / Kendall \(\tau\): rank correlation — what matters for screening
- Top-\(k\) recall: of the true top-\(k\) materials, how many are in the model’s top-\(k\)?
- For discovery, ranking and recall outweigh absolute accuracy
§9 · Case Studies
32. Case studies in one slide
Bandgap
- Intensive \(\Rightarrow\) mean readout
- Needs deeper / global context (MEGNet wins)
- MAE \(\approx 0.3\) eV on MP
Formation energy
- Extensive (per cell) \(\Rightarrow\) sum readout, then \(/N\)
- CGCNN / MEGNet MAE \(\approx 0.03\) eV/atom — near DFT precision
Elasticity
- Sparse data (\(\sim 10^4\) entries)
- Transfer learning: pre-train on energy, fine-tune on \(C_{ij}\)
- Tensor target \(\Rightarrow\) favors equivariant GNNs
§10 · Bridges and Limits
33. Recap to Unit 6 (local atomic environments)
- Each MP layer = one iterative refinement of a local atomic environment descriptor
- Layer \(\ell\) “sees” atoms within graph distance \(\ell\) — analogous to atomic neighbourhood radius
- This is exactly the picture U6 (last week) built with SOAP / ACSF / ACE, which can now be re-read as fixed-feature MP layers — today we make the feature map learnable
34. Bridge to Units 9 and 9
- U9 (NN potentials): trains GNN energies + autograd forces on MD trajectories — equivariance becomes essential
- U10 (representation learning): discards the property head; the GNN encoder produces a latent \(\mathbf{z}\) for self-supervised pre-training, contrastive learning, generative models
35. The limit of GNNs
- GNNs are interpolators trained on DFT data — they inherit DFT’s biases (XC functional, smearing, \(k\)-grid)
- They cannot discover physics absent from the training set (e.g. unconventional superconductivity)
- Long-range Coulomb / dispersion: distance cutoff misses both unless explicit terms are added
- Symmetry-breaking phases (charge density waves, magnetism) require labels that capture them
§11 · Exercises
36. Exercise overview
- Task 1: Build a periodic crystal graph from a CIF using
pymatgen+torch_geometric. Verify edge count under PBC. - Task 2: Train a 3-layer CGCNN on \(\sim 10^4\) ABX\(_3\) perovskites; report MAE, Spearman \(\rho\), top-100 recall for formation energy.
- Task 3: Ablation — sweep \(r_c \in \{4, 6, 8\}\) Å and depth \(L \in \{2, 3, 4, 5\}\). Plot error vs depth; identify over-smoothing.
- Task 4: Failure analysis — inspect 5 worst-error samples. Are they rare chemistries, large cells, or shortcut victims?
§12 · Wrap-Up
37. Unit 7 — Key Takeaways
- A crystal graph \(G=(V,E)\) encodes atoms, bonds, and periodic images via \(\vec{r}_{ij}(\vec{n}) = \vec{r}_j + \mathbf{T}\vec{n} - \vec{r}_i\)
- PBC with the minimum-image convention is mandatory — naive cutoffs disconnect small cells
- Gaussian RBF turns distances into a smooth, differentiable basis that supports autograd forces
- Invariance for scalar targets, equivariance for vector / tensor targets — forces require the latter
- Generic message passing \(h_i^{(\ell+1)} = U(h_i^{(\ell)}, \mathrm{AGG}_j M(h_i^{(\ell)},h_j^{(\ell)},e_{ij}))\) instantiates CGCNN, MEGNet, SchNet
- Sum readout for extensive targets, mean for intensive — never mix them
- Depth \(L = 3\)–\(4\) + skip connections avoids over-smoothing
- Always benchmark against mean / composition-only / SOAP baselines, and report rank metrics for screening
38. Looking back to Unit 6 and forward to Unit 9
- U6 (last week) made local atomic environments explicit: ACSF, SOAP, ACE
- Today’s MP layers are the learned version of those fixed many-body descriptors — same locality, different aggregation
- Next stop: Unit 9 (NN interatomic potentials) — apply today’s GNN machinery + autograd to predict forces on MD trajectories
39. Example notebook
Week 5: Crystal graphs + CGCNN — ABX\(_3\) perovskites
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References
© Philipp Pelz - Materials Genomics