Free energy reconstruction methods such as Gaussian Process Regression (GPR) require Jacobians of the collective variables (CVs), a bottleneck that restricts the use of complex or machine-learned CVs. We introduce a neural network surrogate framework that learns CVs directly from Cartesian coordinates and uses automatic differentiation to provide Jacobians, bypassing analytical forms. On an MgCl2 ion-pairing system, our method achieved high accuracy for both a simple distance CV and a complex coordination-number CV. Moreover, Jacobian errors also followed a near-Gaussian distribution, making them suitable for GPR pipelines. This framework enables gradient-based free energy methods to incorporate complex and machine-learned CVs, broadening the scope of biochemistry and materials simulations.

翻译：诸如高斯过程回归（GPR）等自由能重构方法需要集体变量（CVs）的雅可比矩阵，这一瓶颈限制了复杂或机器学习CVs的使用。我们引入了一种神经网络代理框架，该框架直接从笛卡尔坐标学习CVs，并利用自动微分提供雅可比矩阵，从而绕过了解析形式。在MgCl2离子配对系统中，我们的方法对于简单的距离CV和复杂的配位数CV均实现了高精度。此外，雅可比矩阵误差也遵循近似高斯分布，使其适用于GPR流程。该框架使得基于梯度的自由能方法能够整合复杂及机器学习的CVs，从而拓宽了生物化学和材料模拟的应用范围。