Problem: Explain what an MLIP is.
Solution: A better name might have been MLPES (machine-learned potential energy surface), but MLIP (machine-learned interatomic potential) is the standard term (it should not be confused with e.g. a pairwise interaction potential like the Lennard-Jones potential).
That is, within the Born-Oppenheimer approximation, the ground state PES is just an effective potential \(V(\mathbf x_1,…,\mathbf x_N)\) defined on the nuclear configuration space \((\mathbf x_1,…,\mathbf x_N)\in\mathbf R^{3N}\) on which the nuclei dynamics are adiabatic. Treating the nuclei as classical point masses, it follows that the \(i^{\text{th}}\) nucleus experiences a conservative force field \(\mathbf F_i(\mathbf x_1,…,\mathbf x_N)=-\partial V/\partial\mathbf x_i\).
An MLIP is then just some parametric neural network ansatz seeking to approximate \(\hat V(\mathbf x_1,…,\mathbf x_N|\boldsymbol{\theta})\approx V(\mathbf x_1,…,\mathbf x_N)\). The corresponding MLFF (machine-learned force field) is thus \(\hat{\mathbf F}_i(\mathbf x_1,…,\mathbf x_N|\boldsymbol{\theta}):=-\partial\hat{V}/\partial\mathbf x_i\) (note: the nuclear charges \(Z_1,…,Z_N\in\mathbf Z^+\) of the \(N\) nuclei are also considered inputs to the MLIP).
To be more strict, not just any estimator \(\hat V(\mathbf x_1,…,\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})\) counts as an MLIP. There are certain fundamental requirements (the LAS trio: locality, analyticity, symmetry) that must be fulfilled:
- Locality: \(\hat V(\mathbf x_1,…,\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})=\sum_{i=1}^N\hat V_i\) can be decomposed as a sum of local potentials experienced by each of the \(N\) nuclei, that is \(\hat{V}_i\) depends only on the nuclei \(\{(\mathbf x_j,Z_j):|\mathbf x_j-\mathbf x_i|<r^*\}\) in an \(r^*\)-ball neighbourhood of \(\mathbf x_i\) (strictly speaking, the locality assumption may be relaxed in some cases if long-range electrostatics matters). By the triangle inequality, a corollary of this is that \(|\mathbf x_i-\mathbf x_j|>2r^*\Rightarrow\partial\hat{\mathbf F}_i/\partial\mathbf x_j=0\).
- Analyticity: \(\hat V(\mathbf x_1,…,\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})\) is smooth with respect to the nuclear coordinates \((\mathbf x_1,…,\mathbf x_N)\in\mathbf R^{3N}\), so in particular the forces \(\mathbf F_i\) actually exist!
- Symmetry: the usual Noetherian symmetries \(\hat V(\mathbf x_1+\Delta\mathbf x,…,\mathbf x_N+\Delta\mathbf x|Z_1,…,Z_N,\boldsymbol{\theta})=\hat V(R\mathbf x_1,…,R\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})=\hat V(\mathbf x_1,…,\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})\) for \(\Delta\mathbf x\in\mathbf R^3\) and \(R\in SO(3)\) and discrete symmetries \(\hat V(\mathbf x_{\sigma(1)},…,\mathbf x_{\sigma(N)}|Z_{\sigma(1)},…,Z_{\sigma(N)},\boldsymbol{\theta})=\hat V(-\mathbf x_1,…,-\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})=\hat V(\mathbf x_1,…,\mathbf x_N|Z_1,…,Z_N,\boldsymbol{\theta})\) for \(\sigma\in S_N\).
Problem: One prominent example of an MLIP is MACE, developed by the group of Gábor Csányi at the University of Cambridge. Walk through a single forward pass of MACE at inference time. Then, explain how MACE is trained.
Solution: MACE is an MLIP built on a variant of the MPNN architecture subclass of GNNs. In addition to the standard LAS inductive biases of most MLIPs, MACE itself introduces several further innovations.