Machine-Learning Interatomic Potential

Fixing the head to a physical form turns symmetry and conservation from properties to be trained into properties that hold by construction.

A Machine-Learning Interatomic Potential (MLIP) learns the map from an atomic configuration to its energy and forces. The informative move lies not in the learning but in the fixed head into which the learned part is wired. The total energy is imposed as a sum of local atomic contributions, , where each reads only a local environment through a symmetry-invariant (or $E(3)$-equivariant) descriptor of the neighbours within a cutoff. Euclidean symmetry (rotation, translation, reflection) and permutation of like atoms are therefore respected exactly for every configuration, rather than approximately over a training distribution, because the functional form is a scalar energy over invariant inputs.

Forces are not a second output head to be fit; they are the exact gradient of the same scalar , taken by reverse-mode automatic differentiation of the network with respect to the atomic coordinates. The same computational graph as backpropagation returns analytic forces rather than finite differences, and differentiating through the fixed sum makes the force field conservative by construction. Energy is a genuine potential, so a closed dynamical loop returns zero net work and long molecular-dynamics trajectories do not leak energy. This is the taxonomy thesis in miniature. A hard constraint restricts the space of representable maps before any data is seen, so the learner cannot represent a non-conservative or non-invariant potential even if the loss would reward it.

Hard constraint vs. soft penalty

A soft version would add a regularising term penalising rotational or energy-conservation violations, leaving the unconstrained map in the hypothesis space and hoping training suppresses it. The fixed head instead removes those maps entirely. Symmetry and conservation are properties of the form, exact off-distribution and at any scale, not artefacts of the fit.

Attribution. The local-energy decomposition , realised as high-dimensional neural-network potentials built from per-atom contributions over symmetry functions, is due to Behler and Parrinello. The -equivariant message-passing lineage that makes the descriptors equivariant rather than merely invariant runs through Batzner et al. (NequIP) and Batatia et al. (MACE). The same equivariance is the subject of geometric-deep-learning; adjacent hard-constraint formulations include topological-deep-learning and physics-informed-neural-network.