Towards Efficient Instanton Rate Calculations using Machine Learning Surrogates

We develop a Gaussian process regression enhanced line integral string method to accelerate ring polymer instanton calculations of tunneling rates in molecular proton transfer reactions. By exploiting uncertainty estimates from the surrogate modeling, we show that the number of force evaluations required to converge an instanton path becomes effectively independent of the number of beads used to discretize the pathway. To reduce the computational overhead associated with training, particularly when Hessian information is included, we implement an efficient training strategy by combining physical GPR prior, Hessian-free GPR training and graphics processing unit accelerated black box matrix matrix multiplication, achieving an order of magnitude speedups relative to standard implementations. For rate calculations, we introduce a selective Hessian training strategy that distinguishes flexible modes strongly coupled to the transferring proton from more rigid modes weakly coupled to the reaction coordinate. This enables the construction of accurate surrogate potential energy surfaces with reduced Hessian evaluations. We apply both cubic spline interpolation method and Gaussian Process Regression to approximate the instanton rate for the prototypical systems, malonaldehyde, Z-3-aminopropenal and 7,9-dinitro-10-hydroxybenzo[h]quinoline. In our numerical test, the spline interpolation emerges as a simple and computationally efficient approach for the instanton rate calculations.