Concepts and Terminology#

This page provides concise definitions of the main physical, computational, and adversarial concepts used throughout the project.

Core concepts#

Density functional theory#

Density functional theory (DFT) is an electronic-structure method used to calculate properties such as atomic energies and forces. DFT provides much of the reference data used to train MLFFs, but its computational cost limits the size and duration of practical simulations.

Machine learning force field#

A machine learning force field (MLFF) learns energies and forces from reference electronic-structure calculations. Once trained, it can evaluate atomic structures much faster than running DFT directly. An MLFF remains an approximation and may become unreliable for configurations outside its training distribution.

Training distribution#

The training distribution is the range of chemistries, geometries, energies, and forces represented in a model’s training data. Predictions are generally most trustworthy near this distribution and less certain for unfamiliar atomic environments.

Relaxation#

Atomic relaxation#

A relaxation repeatedly evaluates a structure’s energy and forces and uses the L-BFGS optimizer to update the atomic positions. The process stops when the maximum atomic force satisfies the convergence threshold or when the configured step limit is reached.

Relaxation finds a local minimum under the selected MLFF. It does not, by itself, prove that the resulting structure is physically correct or agrees with DFT.

First relaxation#

The first relaxation converts the input structure into a model-specific reference configuration. Forces, energy, geometry, symmetry, and topology are recorded from this reference before an attack is applied.

Second relaxation#

The second relaxation begins from the perturbed structure and tests recovery. Its final configuration is compared with the first-relaxation reference. A run reaching the 600-step limit without satisfying the force criterion is recorded as non-converged, not fully recovered.

Relaxation trajectory#

A trajectory stores the sequence of atomic configurations produced during relaxation. It can be used to examine how energy, forces, volume, and atomic positions evolve at every optimizer step.

Models#

MACE#

Message Passing Atomic Cluster Expansion (MACE) uses higher-order equivariant message passing to predict atomic energies and forces. Atoms are represented as graph nodes, neighbour relationships as edges, and geometric information is combined through equivariant tensor products. Equivariance ensures that rotating the input produces the corresponding rotation of vector-valued predictions.

The project-specific MACE model is trained using LiCOHPF data.

See MACE.

MACE-MH#

MACE-MH is a pretrained MACE-family model intended to cover a broader range of materials and atomic environments than the project-specific MACE model.

See MACE-MH.

UMA#

Meta’s Universal Model for Atoms (UMA) is a general-purpose pretrained model developed from a large and chemically diverse collection of atomic structures. Its predictions can be conditioned by task, charge, and spin settings.

See UMA.

CHGNet#

Crystal Hamiltonian Graph Neural Network (CHGNet) is a graph-based MLFF designed for crystalline materials. It uses message passing between atoms and their neighbours to predict energies, forces, stresses, and related material properties.

See CHGNet.

MTP#

A Moment Tensor Potential (MTP) represents atomic environments using moment-tensor descriptors and fits an efficient parameterized energy model. MTP calculations are typically computationally compact. The MTP used in this project is trained using LiCOHPF data.

See MTP.

Random seeds#

A random seed initializes stochastic operations so that an experiment can be repeated. In this project, seeds control random components where present; they do not retrain or randomly reinitialize the fixed pretrained model weights. Comparing seeds 42–46 tests whether reported conclusions are stable to stochastic variation.

Adversarial methods#

Adversarial attack#

An adversarial attack deliberately changes atomic coordinates using the gradient of a selected objective. The goal is to find small, controlled perturbations that expose unstable or unreliable model behaviour.

These are white-box attacks because they require access to gradients from the evaluated model.

Attack budget#

The attack budget, \(\epsilon\), limits the permitted coordinate perturbation. This project additionally reports epsilon as a percentage of the structure’s minimum lattice-vector length to support comparison between structures of different sizes.

FGSM#

The Fast Gradient Sign Method (FGSM) applies one coordinate update in the sign direction of the objective gradient. It provides a computationally inexpensive single-step attack.

See FGSM.

I-FGSM#

The Iterative Fast Gradient Sign Method (I-FGSM) divides the attack into multiple smaller gradient-sign updates. Intermediate configurations remain restricted by the selected attack budget.

See I-FGSM.

PGD#

Projected Gradient Descent (PGD) applies repeated gradient-based updates and projects the result back into the allowed perturbation region after each step. It is a strong first-order white-box attack, but it should not be described as universally strongest for every model and objective.

See PGD.

Non-adversarial baseline#

Contour exploration#

Contour exploration follows an isoenergy path, providing a non-adversarial baseline that moves through a targeted approximately constant-energy region instead of following an adversarial gradient. It helps determine whether failures are caused specifically by adversarial direction selection or merely by moving atoms away from equilibrium.

See Contour exploration.