Known transformations are not data to relearn.
Rotating an atomic environment should rotate vector features predictably while leaving scalar energy unchanged.
An interactive field guide to NequIP and its descendants
Teach the model
how space behaves.
NequIP’s durable idea is deceptively simple: do not erase geometry at the input. Carry scalars, vectors, and higher tensors through the network with the right transformation laws; take an invariant only when physics asks for one.
invariant outputE = Σᵢ EᵢFᵢ = −∇ᵢE
ℓ = 0 scalars
ℓ = 1 vectors
ℓ ≥ 2 tensors
typed geometry inside · scalar energy outside
One energy surface disciplines every force.
Differentiation ties force components together and rules out non-integrable vector fields by construction.
The frontier is architecture × data × kernels.
Today’s leaders inherit the blueprint but train on up to roughly 113 million frames and optimize the tensor algebra aggressively.
Prompted by the July Gradient Dissent discussion
“What is the genuinely new mathematical object—and can we make it runnable?” The tutorial answers with group actions, angular rank, conservative forces, and a benchmark game.
At 42 degrees, the equivariance error is 5.55e-17 and invariant-energy drift is 0.000000.
live group action · a 2D slice of E(3)
Experiment 01
Learn once. Rotate everywhere.
The orange hidden vector follows the atoms. Its length can feed an invariant energy head, while its direction remains available for later tensor operations. A coordinate-bound head instead changes its answer when the camera turns.
equivariance error< 10⁻¹²‖m(gx) − g·m(x)‖
invariant energy drift0.000000uses ‖m‖²
coordinate-bound drift+0.048uses mₓ²
hidden vector norm0.342orientation kept, norm invariant
Experiment 02
How much angular structure should the model keep?
Fit the same noisy triatomic energy with three closed-form ridge models. All see bond lengths; richer bases progressively see cos θ and cos² θ. The progression is an analogy to retaining angular information—not a mapping from powers of cos θ to NequIP irreps or a reimplementation of its network.
toy target
E = radial terms + 0.92(cos θ + 0.38)²test RMSE ratio22.5×
radial-only error ÷ quadratic-angular error on 600 held-out structures
NequIP ablationℓ = 0 → 1 changes the learning curve
In the paper’s water study, ℓ = 1, 2, 3 beat ℓ = 0 at every training-set size. Higher ℓ mainly shifted the curve downward after the ℓ = 0 → 1 jump.
With curl 0.22, conservative maximum drift is3.95e-5 and direct-force maximum drift is6.17e+0.
energy-gradient force direct force + curl
Experiment 03
Equivariance is not conservation.
A direct vector force can rotate correctly in 2D and still do nonzero work around a closed loop. NequIP instead predicts one scalar energy and differentiates it: F = −∇E. Turn up the curl contamination and watch the orange trajectory gain or lose energy.
gradient max drift3.95e-5finite-step integration error
direct-force max drift6.17e+0integrator + nonzero curl
closed-loop work1.382∮ F·dr on a unit circle
final energy change+6.166direct-force trajectory
The model has F1 0.526. Its 56-candidate validation budget finds 31 stable candidates at55.4 percent precision.
DFT-stable selected false-positive selected
Experiment 04
A low error is not yet a discovery machine.
Only about 15% of the benchmark pool is stable. Perturb hull energies near zero, then spend a limited “DFT budget” on the candidates the model ranks most stable. Accuracy can look comfortable while false positives consume expensive relaxations.
accuracy85.0%
F10.526precision–recall balance
budget precision55.4%31 stable found
budget DAF4.08×precision ÷ base rate
Genealogy, not a victory parade
NequIP is the hinge between tensor-field theory and usable interatomic potentials.
Its descendants did not simply stack more layers. They attacked different bottlenecks: explicit many-body order, attention, SO(2) reductions, smooth cutoffs, enormous training corpora, and fused kernels. Click the spine to follow what each generation kept.
ACE branchNequIP-like tensors → MACE → TACE → TECE
attention branchTFN/e3nn → Equiformer → V2/eSCN → V3
systems branchCG products → FlashTP/cuEquivariance → EquFlashV2
efficient SO(2) branchdirectional equivariance → eSCN/eSEN → DPA4
NequIP’s synthesis
Keep tensors equivariant inside a local GNN, read out invariant energy, and differentiate it for forces.
What survives downstreamA practical symmetry + locality + conservation blueprint for atomistic learning.
Open primary source ↗Live formula · audited 2026-07-28 · data commit 2026-07-26
There is no single “leader.” Choose the question first.
EquiformerV3 leads unique-prototype discovery F1. TECE leads the default cross-task CPS v1. DPA4 leads CPS inside the benchmark’s open, MP-anchored compliant cohort. Reweight the CPS components below and watch the ordering move.
| rank | model / architectural family | score | F1 ↑ | RMSD ↓ | κSRME ↓ | params |
|---|---|---|---|---|---|---|
| 1 | TECE-OAM-RRA-1.0ACE + edge clusters + rotary attention · Jul 2026 | 0.9076 | 0.929 | 0.0575 | 0.0927 | 221.9M |
| 2 | EquFlashV2Clebsch–Gordan tensor products · Jun 2026 | 0.9072 | 0.929 | 0.0577 | 0.0941 | 44.9M |
| 3 | EquiformerV3+DeNS-OAMSE(3) graph-attention Transformer · Apr 2026 | 0.9023 | 0.931 | 0.0595 | 0.1178 | 30.3M |
| 4 | GRACE-3L-OAM-Llinear ACE equivariant potential · Jul 2026 | 0.8999 | 0.925 | 0.0575 | 0.1211 | 42.1M |
| 5 | PET-OAM-XLpoint-edge Transformer · Jan 2026 | 0.8984 | 0.924 | 0.0596 | 0.1192 | 730M |
| 6 | DPA-4.0.1-Pro-MPtrjEMFA SO(2) convolution · Jun 2026 | 0.8404 | 0.857 | 0.0687 | 0.2114 | 22.8M |
| 7 | EquiformerV3+DeNS-MPSE(3) graph-attention Transformer · Apr 2026 | 0.8297 | 0.863 | 0.0701 | 0.2753 | 30.3M |
CPS v1 default = 0.5F1 + 0.1·clip(1 − RMSD/0.15) + 0.4·(1 − κSRME/2). The sliders normalize to their displayed total. Data, training cost, inference speed, MD, and diatomic behavior are outside this score.
Research report · audited 2026-07-28
The compact reading
The paper’s result is best read as a controlled claim about internal representations, followed by a compelling—but not perfectly controlled—portfolio of atomistic demonstrations.
Fixed tensor algebra, learned radial chemistry
Each edge filter factors into a learned radial function R(r) and a spherical harmonic Yℓm(r̂). Clebsch–Gordan coefficients combine neighbor features and filters along only symmetry-allowed paths. Learnable weights live in invariant radial functions and channel mixing.
Sm(ℓ)(r⃗) = R(r)Ym(ℓ)(r̂)The ℓ = 0 versus ℓ > 0 learning curves
Parameter- and feature-controlled scalar networks retained shallow learning slopes, while equivariant networks learned faster. The big qualitative change appeared at ℓ = 0 → 1; higher ranks usually improved the constant factor.
Aspirin force slope: about −0.38 → −0.69133 water/ice frames versus 133,500
NequIP beat the reported older DeepMD model’s force errors with 1,000× fewer frames. Treat this as an impressive systems result, not a clean architecture ablation: the baseline was nonsmooth, energy tradeoffs remained, and frames came from correlated trajectories.
Dynamics made the case scientifically stronger
The paper reproduced glass radial/angular distributions and obtained lithium diffusivity within roughly 9% of AIMD. These tests ask whether small force errors compound into the wrong structure or kinetics.
Locality, long range, and interpretability
Finite cutoffs and depth impose a finite receptive field. The paper leaves open how many-body order emerges, how to add electrostatics/dispersion, and whether latent atomic energies can be interpreted physically.
NequIP became a blueprint, not the final winner
TECE, EquFlashV2, EquiformerV3, and DPA4 keep symmetry-typed features and conservative energies while changing body order, attention, SO(2) reductions, nonlinearities, data scale, and implementation strategy.