Constrained Potential Fitting

The constrained potential fitting process aims to fit a cheap potential to a highly localized region of the potential energy surface of an expensive reference model. This approach enables fitting a cheap potential which approximates an expensive potential well in a simulation while maintaining low computational complexity and cost.

Selecting Constraints

The fitting process involves two types of constraints: soft constraints and hard constraints.

• Soft Constraints: These are loose matching conditions that allow flexibility in the fit, suitable for general properties of the potential.

• Hard Constraints: These are tight matching conditions, often used for critical properties like elastic constants. For example, ensuring the seamless matching of long-range elastic stress fields between the cheap and expensive potentials in solid-state simulations.

The constraints are encoded as design matrices, denoted as \(A_H\) for hard constraints and \(A_S\) for soft constraints, with corresponding fitting data vectors \(y_H\) and \(y_S\).

Constrained Fitting

The constrained optimization problem can be expressed as:

\[\min_{\mathbf{c}, ||\mathbf{A}_\mathrm{H} \mathbf{c} - \mathbf{y}_{\mathrm{H}}||^{2} < \alpha} \left(||\mathbf{y}_{\mathrm{S}} - \mathbf{A}_\mathrm{S}\mathbf{c}||^{2} \right)\]

Where:

• \(c\) represents the potential parameters.

• \(A_H\) and \(A_S\) are the design matrices for the hard and soft constraints, respectively.

• \(y_H\) and \(y_S\) are the target data vectors for the hard and soft constraints.

• \(\alpha\) is the maximum allowable error for the hard constraint data.

The problem can be solved using an augmented Lagrangian approach, where the Lagrange multiplier \(\lambda\) enforces the hard constraint. The augmented Lagrangian function is:

\[\mathcal{L}(\mathbf{c}, \lambda) = ||\mathbf{y}_{\mathrm{S}} - \mathbf{A}_\mathrm{S}\mathbf{c}||^{2} + \lambda (||\mathbf{y}_{\mathrm{H}} - \mathbf{A}_\mathrm{H} \mathbf{c}||^{2} - \alpha)\]

The optimal potential parameters c are found by minimizing this objective function with respect to c, with the constraint controlled by the Lagrange multiplier \(\lambda\).

The solution to this problem requires finding the appropriate value of \(\lambda\) such that the fit lies on the boundary of the hard constraint subspace, i.e.,

\[||\mathbf{A}_\mathrm{H} \mathbf{c} - \mathbf{y}_{\mathrm{H}}||^{2} = \alpha\]

The constrained fitting process can be performed iteratively by adjusting the value of lambda, starting with a logarithmic search to satisfy the hard constraint, followed by interval bisection to refine the solution.