How to differentiate collective variables in free energy codes: Computer-algebra code generation and automatic differentiation

Abstract

The proper choice of collective variables (CVs) is central to biased-sampling free energy reconstruction methods in molecular dynamics simulations. The PLUMED 2 library, for instance, provides several sophisticated CV choices, implemented in a C++ framework; however, developing new CVs is still time consuming due to the need to provide code for the analytical derivatives of all functions with respect to atomic coordinates. We present two solutions to this problem, namely (a) symbolic differentiation and code generation, and (b) automatic code differentiation, in both cases leveraging open-source libraries (SymPy and Stan Math, respectively). The two approaches are demonstrated and discussed in detail implementing a realistic example CV, the local radius of curvature of a polymer. Users may use the code as a template to streamline the implementation of their own CVs using high-level constructs and automatic gradient computation. Program summaryProgram Title: Practical approaches to the differentiation of collective variables in free energy codes: computer-algebra code generation and automatic differentiationProgram Files doi:http://dx.doi.org/10.17632/r4r67bvkdn.1Licensing provisions: GNU Lesser General Public License Version 3 (LGPL-3)Programming languages: C++, PythonNature of problem: The C++ implementation of collective variables (CVs, functions of atomic coordinates to be used in biased sampling applications) in biasing libraries for atomistic simulations, such as PLUMED [1], requires computation of both the variable and its gradient with respect to the atomic coordinates; coding and testing the analytical derivatives complicate the implementation of new CVs.Solution method: The paper shows two approaches to automate the computation of CV gradients, namely, symbolic differentiation with code generation and automatic code differentiation, demonstrating their implementation entirely with open-source software (respectively, SymPy and the Stan Math Library).Additional comments: The paper’s accompanying code serves as an example and template for the methods described in the paper; it is distributed as the two modules curvature_codegen and curvature_autodiff integrated in PLUMED 2’s source tree; the latest version is available at https://github.com/tonigi/plumed2-automatic-gradients .[1] Tribello GA, Bonomi M, Branduardi D, Camilloni C, Bussi G. PLUMED 2: New feathers for an old bird. Computer Physics Communications. 2014 Feb;185(2):604-13.