Chapter 9 — Machine Learning Interatomic Potentials¶
For seventy years the practitioner of atomistic simulation has faced an uncomfortable choice. Quantum-mechanical methods such as density functional theory (Chapter 5) deliver chemically accurate energies and forces, but the \(O(N^3)\) scaling and large prefactor limit them to systems of a few hundred atoms and timescales of picoseconds. Classical force fields (Chapter 7), written as hand-tuned functional forms in interatomic distances and angles, scale linearly and run on a laptop, but they capture only a narrow slice of chemistry: change the bonding environment and the parameters break.
The gap between these two regimes — call it the accuracy–cost gap — has shaped what materials scientists could and could not study. Reactive chemistry in a million-atom electrolyte, the nucleation of a solid from its melt, the diffusion of a defect over microseconds: all sat in the no-man's land between DFT and Lennard-Jones.
Machine learning interatomic potentials (MLIPs) close that gap. The idea is brutally simple: generate a few thousand DFT calculations on small reference configurations, fit a flexible regression model to the resulting energy-force-stress data, and use the fitted model in place of DFT inside your molecular dynamics or geometry-optimisation engine. If the model generalises, you obtain DFT-quality forces at the cost of evaluating a few million floating-point operations per atom — roughly four orders of magnitude cheaper than the underlying DFT.
What makes this work is not the universal-approximator theorem (regression on a sufficiently rich basis can fit anything; the question is whether it generalises) but a careful encoding of physics. An interatomic potential is not an arbitrary function of \(3N\) coordinates: it is invariant under translation, rotation, and permutation of identical atoms; it must be smooth so that forces are continuous; it must be local so that energy decomposes into atomic contributions that depend only on a neighbourhood. Every successful MLIP architecture — Behler–Parrinello neural networks, Gaussian Approximation Potentials, SchNet, NequIP, MACE — bakes these constraints into its representation. Models that ignore them require orders of magnitude more training data, and still extrapolate badly.
This chapter develops the field from first principles. We begin with the motivation: why MLIPs exist and what they can do (Section 9.1). We then catalogue the symmetries any potential must respect, and distinguish invariant features (scalars under rotation) from equivariant features (vectors and tensors that transform predictably) — a distinction that separates first-generation MLIPs from the modern equivariant networks (Section 9.2). With those constraints in hand we survey the three descriptor families that dominate the literature: Behler–Parrinello symmetry functions, the Smooth Overlap of Atomic Positions (SOAP), and the Atomic Cluster Expansion (ACE) (Section 9.3).
Section 9.4 turns to two complete architectures built on these descriptors:
Behler–Parrinello neural networks and Gaussian Approximation Potentials
(GAP). These are the workhorses of the 2010s and remain competitive on many
problems. Section 9.5 introduces the equivariant revolution — NequIP and
MACE — and explains why operating on irreducible representations of
\(\mathrm{O}(3)\) yields models that learn roughly twenty times faster than
their invariant predecessors. Section 9.6 is hands-on: we install
mace-torch, train a MACE potential on a water dataset, and wire the
trained model into ASE for molecular dynamics. The chapter closes with
eight exercises that take the reader from deriving forces by autograd to
fitting and validating their own potential.
Prerequisites. You should be comfortable with the DFT workflow of Chapter 5 (we use it to generate labels), with the MD integrators of Chapter 7 (we will feed the MLIP into them), and with the statistical mechanics of Chapter 8 (we use ensembles and free energies as validation tools). On the machine-learning side we assume only that you have encountered a feed-forward neural network: every architecture in this chapter is built up explicitly from linear layers, nonlinearities, and a small amount of group theory which we develop as needed.
What to expect. By the end of the chapter you will have:
- a precise statement of the symmetries any interatomic potential must satisfy, and the tools to check them;
- a working implementation of a Behler–Parrinello \(G^2\) descriptor;
- the ability to read and critique papers in the SOAP/ACE/MACE family;
- a trained MACE potential for liquid water with a force MAE of roughly \(30\,\mathrm{meV}/\text{\AA}\);
- and the connections to Chapter 10 (graph neural networks generalise the message-passing structure introduced here), Chapter 11 (active learning closes the loop by querying DFT only where the MLIP is uncertain), and Chapter 12 (foundation models such as MACE-MP-0 amortise the training cost across all of materials chemistry).
The chapter is opinionated. We have chosen MACE as the working example because, at the time of writing (2026), it represents the best trade-off between accuracy, data-efficiency, and inference cost for a single researcher with one GPU. The principles transfer directly to NequIP, Allegro, SevenNet, and the other equivariant architectures you will encounter in the literature.