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International Symposium on Molecular Spectroscopy (ISMS)

UTILIZATION OF NEURAL NETWORKS FOR GPU-ACCELERATED DIFFUSION MONTE CARLO FOR VIBRATIONAL PROBLEMS

Fenris Lu, Anne B. McCoy, Ryan J. DiRisio

Published Jun 22, 2021• 0 citations
Publisher

Abstract

In recent years, there has been increasing interest in combining machine learning with computational chemistry. Many machine learning applications in theoretical chemistry have focused on the development of potential energy surfaces (PES) based on \emph{ab initio} data points\footnote{Manzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665}. Another computational resource that has gained popularity in recent years is the use of Graphics Processing Units (GPUs) to accelerate calculations such as molecular dynamics simulations \footnote{Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25}. In this work, we take advantage of the knowledge gained from both of these areas, and apply these approaches to the evaluations of the potential energy for Diffusion Monte Carlo (DMC) calculations. Diffusion Monte Carlo is a stochastic simulation approach through which an ensemble of localized functions (walkers) converges to the ground state vibrational wave function at long propagation times. This is accomplished through the ensemble randomly sampling a potential energy surface. These calculations require on the order of 10$^8$ to 10$^10$ potential energy evaluations, making the call to a potential energy surface the most expensive part of these calculations. To this end, we have developed a general technique for “learning” already existing potential energy surfaces using neural networks, which can then easily be evaluated on GPUs using TensorFlow. The technique entails running a small-scale DMC simulation to collect training data, coordinates and energies, and using this data to fit a subspace of the potential energy surface that is relevant for the DMC simulations. We find that using the neural network potential energy surface maintains the fidelity of the wave function while also enabling the use of ensemble sizes orders of magnitude larger than previously used for the same amount of wall time. We test this method on small systems such as \chem{CH5+}, \chem{H2O}, and water dimer.

DOI
10.15278/isms.2021.wk01

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In recent years, there has been increasing interest in combining machine learning with computational chemistry. Many machine learning applications in theoretical chemistry have focused on the development of potential energy surfaces (PES) based on \emph{ab initio} data points\footnote{Manzhos, S.; Carrington, T. Chem. Rev. Article ASAP. https://doi.org/10.1021/acs.chemrev.0c00665}. Another computational resource that has gained popularity in recent years is the use of Graphics Processing Units (GPUs) to accelerate calculations such as molecular dynamics simulations \footnote{Abraham, M. J.; Murtola, T.; Schulz, R.; Páll, S.; Smith, J. C.; Hess, B.; Lindahl, E. SoftwareX 2015, 1–2, 19–25}. In this work, we take advantage of the knowledge gained from both of these areas, and apply these approaches to the evaluations of the potential energy for Diffusion Monte Carlo (DMC) calculations. Diffusion Monte Carlo is a stochastic simulation approach through which an ensemble of localized functions (walkers) converges to the ground state vibrational wave function at long propagation times. This is accomplished through the ensemble randomly sampling a potential energy surface. These calculations require on the order of 108^88 to 1010^1010 potential energy evaluations, making the call to a potential energy surface the most expensive part of these calculations. To this end, we have developed a general technique for “learning” already existing potential energy surfaces using neural networks, which can then easily be evaluated on GPUs using TensorFlow. The technique entails running a small-scale DMC simulation to collect training data, coordinates and energies, and using this data to fit a subspace of the potential energy surface that is relevant for the DMC simulations. We find that using the neural network potential energy surface maintains the fidelity of the wave function while also enabling the use of ensemble sizes orders of magnitude larger than previously used for the same amount of wall time. We test this method on small systems such as \chem{CH5+}, \chem{H2O}, and water dimer.
Diffusion Monte Carlo (DMC) is a technique for obtaining the ground-state solution to the vibrational time-independent Schrödinger equation based on a stochastic sampling of an electronic potential energy surface (PES). Ideally, the electronic energies used in DMC are calculated at the CCSD(T) level of theory. Recently, Tom Miller’s group at Caltech has developed a technique for calculating accurate CCSD(T) corrections to the electronic energy at the cost of a standard Hartree-Fock calculation using Gaussian process regression, a machine learning technique\footnote{Welborn, M. , Cheng, L., Miller, T. M. JCTC. 2018 14 (9), 4772-4779}\footnote{Cheng, L., Kovachki, N. B., Welborn, M., Miller, T. M. JCTC. 2019 15 (12), 6668-6677}. DMC is a well-suited technique to validate these PESs, referred to as MOB-ML surfaces, since DMC samples the entire region of the PES in which the vibrational ground state wave function has amplitude. Despite the speed of these MOB-ML PESs relative to an \emph{ab initio} calculation, calculating the Hartree-Fock energy for every one of thousands of DMC configurations over tens of thousands of time-steps makes the use of these surfaces computationally challenging. To this end, we combined two techniques to bring these calculations into computational feasibility. First, we implemented a DMC algorithm to take advantage of massively parallel high performance computing environments. Second, we use this DMC code to collect training data for a neural network (NN). We use a generic algorithm to train the NN to learn the MOB-ML surface based on the points sampled by the DMC. Using TensorFlow, the evaluation of the NN can then easily be done using graphics processing units (GPUs). We have performed small-scale DMC calculations on \chem{H2O} and \chem{CH5+} using these MOB-ML surfaces, and have also run large-scale NN+MOB-ML DMC simulations to obtain a high resolution ground state vibrational wave function and measure of the zero-point energy. Due to the generality of both the MOB-ML surface generation and the NN fitting workflow, we can easily extend this work to larger systems, such as large ion-water clusters or a Criegee intermediate, where running the DMC with traditional electronic structure would be intractable.
Convergent hierarchies of theories for calculating many-body vibrational ground and excited-state wave functions, such as M{o}ller-Plesset perturbation theory or coupled cluster theory, tend to rely on matrix-algebraic manipulations of large, high-dimensional arrays of anharmonic force constants, tasks which require large amounts of computer storage space and which are very difficult to implement in a parallel-scalable fashion. On the other hand, existing quantum Monte Carlo (QMC) methods for vibrational wave functions tend to lack robust techniques for obtaining excited-state energies, especially for large systems. By exploiting analytical identities for matrix elements of position operators in a harmonic oscillator basis, we have developed stochastic implementations of the size-extensive vibrational self-consistent field (MC-XVSCF) and size-extensive vibrational M{o}ller-Plesset second-order perturbation (MC-XVMP2) theories which do not require storing the potential energy surface (PES). The programmable equations of MC-XVSCF and MC-XVMP2 take the form of a small number of high-dimensional integrals evaluated using Metropolis Monte Carlo techniques. The associated integrands require independent evaluations of only the value, not the derivatives, of the PES at many points, a task which is trivial to parallelize. However, unlike existing vibrational QMC methods, MC-XVSCF and MC-XVMP2 can calculate anharmonic frequencies directly, rather than as a small difference between two noisy total energies, and do not require user-selected coordinates or nodal surfaces. MC-XVSCF and MC-XVMP2 can also directly sample the PES in a given approximation without analytical or grid-based approximations, enabling us to quantify the errors induced by such approximations.
The ground state of second-quantized quantum chemistry Hamiltonians provides access to an important set of chemical properties. Wave functions based on machine-learning architectures have shown promise in approximating these ground states in a variety of physical systems. In this paper, we show how to achieve state-of-the-art energies for molecular Hamiltonians using the the neural network backflow (NNBF) wave function. To accomplish this, we optimize this ansatz with a variant of the deterministic optimization scheme based on selected configuration interaction introduced by Li et al., [J. Chem. Theory Comput. 19, 8156 (2023)], which we find works better than standard Markov chain Monte Carlo sampling. For the molecules we studied, NNBF gives lower energy states than both Coupled Cluster with Single and Double excitations and other neural network quantum states. We systematically explore the role of network size as well as optimization parameters in improving the energy. We find that, while the number of hidden layers and determinants play a minor role in improving the energy, there are significant improvements in the energy from increasing the number of hidden units as well as the batch size used in optimization, with the batch size playing a more important role.
Diffusion Monte Carlo (DMC) based on fixed-node approximation has enjoyed significant developments in the past decades and become one of the go-to methods when accurate ground state energy of molecules and materials is needed. However, the inaccurate nodal structure hinders the application of DMC for more challenging electronic correlation problems. In this work, we apply the neural-network based trial wavefunction in fixed-node DMC, which allows accurate calculations of a broad range of atomic and molecular systems of different electronic characteristics. Our method is superior in both accuracy and efficiency compared to state-of-the-art neural network methods using variational Monte Carlo (VMC). We also introduce an extrapolation scheme based on the empirical linearity between VMC and DMC energies, and significantly improve our binding energy calculation. Overall, this computational framework provides a benchmark for accurate solutions of correlated electronic wavefunction and also sheds light on the chemical understanding of molecules.
Accurate ab initio calculations are of fundamental importance in physics, chemistry, biology, and materials science, which have witnessed rapid development in the last couple of years with the help of machine learning computational techniques such as neural networks. Most of the recent efforts applying neural networks to ab initio calculation have been focusing on the energy of the system. In this study, we take a step forward and look at the interatomic force obtained with neural network wavefunction methods by implementing and testing several commonly used force estimators in variational quantum Monte Carlo (VMC). Our results show that neural network ansatz can improve the calculation of interatomic force upon traditional VMC. The relationship between the force error and the quality of the neural network, the contribution of different force terms, and the computational cost of each term is also discussed to provide guidelines for future applications. Our work demonstrates that it is promising to apply neural network wavefunction methods in simulating structures/dynamics of molecules/materials and provide training data for developing accurate force fields.
The potential energy surface (PES) of molecules with respect to their nuclear positions is a primary tool in understanding chemical reactions from first principles. However, obtaining this information is complicated by the fact that sampling a large number of ground states over a high-dimensional PES can require a vast number of state preparations. In this work, we propose using a generative quantum machine learning model to prepare quantum states at arbitrary points on the PES. The model is trained using quantum data consisting of ground-state wavefunctions associated with different classical nuclear coordinates. Our approach uses a classical neural network to convert the nuclear coordinates of a molecule into quantum parameters of a variational quantum circuit. The model is trained using a fidelity loss function to optimize the neural network parameters. We show that gradient evaluation is efficient and numerically demonstrate our method's ability to prepare wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In all cases, we find that a small number of training points are needed to achieve very high overlap with the groundstates in practice. From a theoretical perspective, we further prove limitations on these protocols by showing that if we were able to learn across an avoided crossing using a small number of samples, then we would be able to violate Grover's lower bound. Additionally, we prove lower bounds on the amount of quantum data needed to learn a locally optimal neural network function using arguments from quantum Fisher information. This work further identifies that quantum chemistry can be an important use case for quantum machine learning.