The single architectural distinction that separates 2017-era neural network potentials from today's state of the art: what happens to a network's internal features when you rotate the crystal — and why that determines force accuracy.
Output unchanged under rotation
Output rotates with the input
SchNet 2017 → MACE 2022
Far more accurate forces, fewer training structures
Post 16 established that every neural network potential sums per-atom energy contributions computed from each atom's local environment. The question this post answers is: what mathematical object does the network use to represent that local environment? The answer — scalars only (invariant) versus scalars plus vectors and tensors (equivariant) — turns out to be the single biggest factor separating SchNet-era accuracy from MACE-era accuracy.
1. Two Symmetries, One Crystal
Physically, a crystal's total energy cannot depend on how you've oriented it in space — rotate the whole structure and the energy stays exactly the same. This is rotational invariance of the energy, and it is non-negotiable: any correct potential must respect it. But the forces on each atom are vectors — they point in specific directions in 3D space. Rotate the crystal, and the force vectors must rotate by exactly the same amount. This is rotational equivariance of the forces.
Forces (vector): F(RR) = R · F(R) — equivariant
R = a rotation matrix applied to all atomic positions
RR = the rotated set of positions
Imagine weighing a rotating object on a scale — the weight reading (a scalar) never changes no matter which way the object is turned: that's invariance. Now imagine tracking the direction an arrow on that object points — as you rotate the object, the arrow's direction rotates with it: that's equivariance. A network that only ever computes scalars cannot represent that arrow at all.
2. SchNet: Invariant by Construction
SchNet represents each atom's environment using only interatomic distances — a quantity that is automatically rotation-invariant, since rotating the crystal doesn't change how far apart any two atoms are. Every internal feature SchNet computes is a scalar number per atom. This guarantees the predicted energy is invariant (correct), but it means SchNet must recover the directional information needed for forces entirely through differentiation of distance-based features — it never explicitly represents "which direction".
Every input to the filter W is a scalar distance ‖ri−rj‖.
No directional (vector) information ever enters the network.
3. NequIP and MACE: Equivariant by Construction
NequIP (2022) and MACE (2022) instead pass spherical harmonic representations of the relative position vectors ri−rj through the network — not just their length, but their full directional content, organised by rotation order ℓ (ℓ=0 is a scalar, ℓ=1 transforms like a vector, ℓ=2 like a rank-2 tensor, and so on). Every layer is built from tensor products that combine these multi-order features while provably preserving the correct equivariance at every step.
Yℓ(r̂ij) = spherical harmonics of the unit direction r̂ij
⊗ = tensor product, combined via Clebsch-Gordan coefficients
Output features at ℓ>0 rotate correctly when the input crystal rotates
| Property | SchNet (invariant) | NequIP / MACE (equivariant) |
|---|---|---|
| Internal features | Scalars only (ℓ=0) | Scalars + vectors + tensors (ℓ=0,1,2,…) |
| Input to the network | Interatomic distances only | Full direction vectors ri−rj |
| Force accuracy | Good, but indirect | Substantially better — directional info is explicit |
| Data efficiency | Needs more training structures | Often 10-100× fewer structures for same accuracy |
| Compute cost per step | Lower | Higher (tensor products are expensive) |
4. Why Equivariance Helps Most for Transition Metal Compounds
The energy splitting between t₂g and eg d-orbitals depends on the direction of the ligand field relative to the d-orbital lobes — this is inherently a directional, tensor-like effect (the d-orbitals themselves transform as ℓ=2 spherical harmonics under rotation). An invariant network like SchNet can only learn this indirectly through distance patterns; an equivariant network can represent the directional structure of the ligand field much more naturally, which is part of why MACE-MP-0 performs so well on transition-metal-containing materials straight out of the box.
5. The Architecture Timeline
| Year | Model | Key contribution |
|---|---|---|
| 2007 | Behler-Parrinello | Sum of per-atom NNs; hand-crafted symmetry-function descriptors |
| 2017 | SchNet | Learned continuous-filter convolutions replace hand-crafted descriptors; still fully invariant |
| 2021 | NequIP | First widely-adopted E(3)-equivariant message-passing potential; tensor features throughout |
| 2022 | MACE | Higher-order equivariant many-body messages in a single layer — fewer layers needed for the same expressivity, faster training |
| 2023–24 | MACE-MP-0, CHGNet | Equivariant architectures trained as foundation models on the full Materials Project trajectory database |
6. What MACE Adds on Top of NequIP
Both NequIP and MACE are equivariant, but MACE's key innovation is constructing higher-order many-body features within a single message-passing step, rather than building up many-body correlations gradually over many layers as NequIP does. This makes MACE both more expressive per layer and considerably faster to train — one reason it has become the default starting point for foundation models like MACE-MP-0.
Quick Check
1. What does it mean for a property to be "rotationally equivariant" rather than "rotationally invariant"?
2. Why can SchNet's internal features never directly represent a bond direction?
3. Why might an equivariant network like MACE be especially well-suited to transition-metal d-orbital physics?
My name is Dr Abderrahmane Reggad. I'm a 54 year old and I am a researcher in materials' sciences. 
