### Improved Force Matching Using Bayesâ€™ Theorem and Stretched String Energy

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David Rogers, University of Cincinnati
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Force matching is a widely recognized method for approximating many-body free energy surfaces (PMFs). A major drawback of this method is its reliance on least-squares fitting procedures and arbitrary spline resolution during collection of force averages. This can cause dramatic over-fitting for small sample sizes and subsequently produce unstable coarse-level dynamics. Viewing the the sampled (coarse-level) force as a stochastic variable gives a Bayesian re-interpretation of the fitting procedure. Every observed force has a defined probability of occurring, which gives information on the underlying force distribution. Since we expect functions in the chosen coarse energy expression to be smooth, a prior based on stretched string energy is used to enforce smoothness where no/noisy samples are observed. This gives a method similar in spirit to Bayesian P-splines, with the added advantage of a properly scale- independent prior. Several numerical tests confirm stability with respect to function scale and chosen spline resolution. In addition, the method is shown to match both force surfaces and stochastic force magnitude for common bond, angle, torsion, and pairwise potential surfaces.

**Abstract Author(s): **David M. Rogers and Thomas L. Beck