Machine learning potentials¶

Why settle on a functional form?¶

The potential energy $E_\mathrm{pot}(\mathbf{x})$ has many complex features:

• Number of minima exponential on N

• Multiple transition paths

• Features at different length and energy scales

Approaches to approximate the true potential energy function:

• Quantum-mechanical calculations would be very accurate, but are way too expensive

• The classical forcefields from the previous lectures are cheap but only a crude approximation

• Machine learning can approximate a region of the function close to QM accuracy but with much less expensive function evaluations

Machine learning¶

Regression: Predict the continuous outcome/function value given some new unseen input

Step 1: Collect data (measurements, calculations)

Step 2: Fit model (find best parameters of function)

Step 3: Apply model, e.g. y(2.0)=2.1

Machine-learning (ML) potentials¶

Data: $E_\mathrm{Pot}(\mathbf{x})$ ... potential energy as a function of the atomic coordinates from QM calculations for many relevant conformations (+ optionally atomic forces $\mathbf{f}$)\[0.07cm]

Problem: Learned potential energy would not be permuation, translation, or rotation invariant!

Assumption 1: $E_\mathrm{Pot}(\mathbf{x})$ is approximately additive with respect to the atomic contributions $\mathbf{x}_1, \mathbf{x}_2, ..., \mathbf{x}_N$

Problem: Learned potential energy would not be translation or rotation invariant!

Assumption 2: The atomic contribution $E_{\mathrm{Pot},N}(\mathbf{x})$ can also be learned from $E_{\mathrm{Pot},N}(\mathbf{q_N})$ with $\mathbf{q_N}$ being a descriptor representation of the local environment around atom N

Problem: We must choose/design suitable descriptors

How to design a descriptor¶

Descriptors must be invariant with respect to translation, and invariant or equivariant with respect to rotation.

Simplest choice: Two-body (atom + distance information), or three-body (atom + distance + angle information)

Usually: Many-body representation (all neighboring atoms up to a cutoff) with a radial part and an angular part (usually based on spherical harmonics):

Zubatiuk & Isayev (2021)

Descriptor-based architectures¶

Design of descriptors is still a work in progress, many variants published recently:

• Atom-centered symmetry functions (ACSFs)

• Smooth overlap of atomic positions (SOAP)

• Eigenvalues of the Coulomb matrix

Exemplar architectures:

• GAP: Gaussian process regression on SOAP descriptors

• NeuralIL: Residual neural network on Bessel descriptors

• ANI: Neural network on ACSFs

Learned descriptors¶

So far, we have custom-made the descriptors, relying on expert knowledge which information is important:

Instead, we can learn the descriptor from the data using message passing (= another model):

Evolution of learned descriptors:

• 2-Body (atoms I, J) invariant message passing with scalar features: Very basic learned features

• Multi-body messages including also triples (atoms I, J, K) or more: Can capture more information

• 2-Body (atoms I, J) equivariant message passing with scalar and vectorial features: Can also capture directional information

• Combination of equivariant message-passing with multi-body messages: Currently one of the best models

What about the long-range part?¶

Previous methods only use local descriptors. What about long-range forces though?

Thre is commonly accepted approach yet. Some possibilities:

• Fixed atomic multipoles (charges, dipoles, etc)

• Machine-learning method for the charge density and the derived potential energy

• Project long-range part onto local descriptors

• Ewald-type ML

proceedings.mlr.press/v202/kosmala23a.html

Disadvantages/Challenges¶

The development of machine-learned force fields is far from finished, and many challenges remain. Maybe you want to contribute?

• No functional form means no safety net - predictions can get arbitrarily different from the ground truth. Predictions can thus even have a wrong sign, i.e. be attractive instead of repulsive

• Generalization ability often poor

• High dependence on training set

• Currently promising results after some finetuning, but not off-the-bench