Deep neural network solution of the electronic Schrödinger equation (1909.08423v5)

Abstract: [New and updated results were published in Nature Chemistry, doi:10.1038/s41557-020-0544-y.] The electronic Schr\"odinger equation describes fundamental properties of molecules and materials, but can only be solved analytically for the hydrogen atom. The numerically exact full configuration-interaction method is exponentially expensive in the number of electrons. Quantum Monte Carlo is a possible way out: it scales well to large molecules, can be parallelized, and its accuracy has, as yet, only been limited by the flexibility of the used wave function ansatz. Here we propose PauliNet, a deep-learning wave function ansatz that achieves nearly exact solutions of the electronic Schr\"odinger equation. PauliNet has a multireference Hartree-Fock solution built in as a baseline, incorporates the physics of valid wave functions, and is trained using variational quantum Monte Carlo (VMC). PauliNet outperforms comparable state-of-the-art VMC ansatzes for atoms, diatomic molecules and a strongly-correlated hydrogen chain by a margin and is yet computationally efficient. We anticipate that thanks to the favourable scaling with system size, this method may become a new leading method for highly accurate electronic-strucutre calculations on medium-sized molecular systems.

• The paper introduces PauliNet, a deep-learning framework that integrates quantum physics principles to solve the electronic Schrödinger equation.
• It employs deep Jastrow and backflow transformations with variational quantum Monte Carlo to enhance accuracy and electron correlation capture.
• Numerical experiments show that PauliNet recovers 97% to 99.9% of electron correlation energy, outperforming traditional quantum chemistry methods.

Analyzing Deep Neural Network Solutions to the Electronic Schrödinger Equation

The paper "Deep neural network solution of the electronic Schrödinger equation" by Jan Hermann, Zeno Schätzle, and Frank Noé represents a significant contribution to the computational methods for solving quantum mechanical problems inherent to chemistry and materials science. The authors propose a novel deep-learning framework, named PauliNet, designed to solve the electronic Schrödinger equation—an equation fundamental to understanding molecular and material properties.

Key Contributions

The introduced PauliNet framework embodies a wave function ansatz leveraging deep neural networks (DNNs) to model electronic systems with an accuracy superior to traditional methods like the Hartree-Fock and standard quantum Monte Carlo (QMC) approaches. Key aspects of the PauliNet architecture include:

• Integration of Quantum Physical Knowledge: The architecture encodes essential quantum physics principles, such as Slater determinants, the multi-determinant expansion, and cusp conditions, which are pivotal for representing electronic wave functions accurately.
• Deep Jastrow and Backflow Transformations: The use of DNNs to represent the Jastrow factor and backflow transformations enhances the correlation between electrons and the effectiveness of nodal surface modifications.
• Variational Quantum Monte Carlo Training: PauliNet employs variational QMC that minimizes the total electronic energy by intelligently refining electronic configurations on-the-fly. This training method demonstrates the flexibility of DNNs in optimizing complex quantum systems.

Numerical Results

The numerical experiments conducted on small atomic and molecular systems show that PauliNet successfully recovers between 97% to 99.9% of the electron correlation energy, often matching or exceeding the precision of existing quantum chemistry methods, but with fewer computational resources and input determinants. For instance, computation on the H$_{10}$ hydrogen chain demonstrates PauliNet's capability to handle strong electronic correlations—a challenging aspect for many quantum chemistry techniques.

Implications

The implications of this research are noteworthy. Practically, PauliNet can facilitate highly accurate electronic structure calculations for medium-sized systems at a lower computational cost, potentially enhancing computational chemistry and materials science applications. Theoretically, this research bridges machine learning and quantum chemistry, offering a new paradigm wherein deep neural networks comprehensively analyze and predict electronic structures, possibly mitigating the limitations of existing methods such as the fixed-node errors inherent in QMC.

Future Directions

The trajectory of this research suggests several avenues for future exploration:

• Scalability: While PauliNet shows promise in system size scaling, further research could improve its application to even larger systems and increase the computational throughput, possibly through parallelization and algorithmic optimizations.
• Integration with Other Machine Learning Models: Combining PauliNet with other architectures, such as FermiNet, could lead to hybrid models that capitalize on diverse neural network strengths.
• Exploration Beyond Ground-State Energies: Extending the framework to evaluate excited states and dynamic properties could vastly broaden its applicability within physical sciences.

In summary, this paper introduces a robust integration of deep learning into quantum chemistry, setting a new standard for electronic structure calculations. PauliNet demonstrates the power of neural networks as function approximators in quantum mechanical systems, providing a robust platform for future advancements in both methodology and applied computations.