Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks (1909.02487v3)

Abstract: Given access to accurate solutions of the many-electron Schr\"odinger equation, nearly all chemistry could be derived from first principles. Exact wavefunctions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially-scaling algorithms. The key challenge for many of these algorithms is the choice of wavefunction approximation, or Ansatz, which must trade off between efficiency and accuracy. Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wavefunction Ansatz for spin systems, but problems in electronic structure require wavefunctions that obey Fermi-Dirac statistics. Here we introduce a novel deep learning architecture, the Fermionic Neural Network, as a powerful wavefunction Ansatz for many-electron systems. The Fermionic Neural Network is able to achieve accuracy beyond other variational quantum Monte Carlo Ans\"atze on a variety of atoms and small molecules. Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly-correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry. This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab-initio quantum chemistry methods, opening the possibility of accurate direct optimization of wavefunctions for previously intractable many-electron systems.

• The paper introduces the Fermionic Neural Network (FermiNet) as a novel deep learning approach to solve the many-electron Schrödinger equation ab initio.
• FermiNet achieves high accuracy by using a permutation-equivariant neural network architecture, outperforming traditional methods like CCSD(T) on challenging systems.
• This method offers improved scalability, does not rely on basis sets, and shows promise for applications in larger systems and exploring molecular properties.

Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks: A Comprehensive Analysis

The paper "Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks" presents a significant advancement in quantum chemistry and electronic structure theory by introducing the Fermionic Neural Network (FermiNet) as a wavefunction Ansatz for many-electron systems. The approach aims to improve the accuracy of variational quantum Monte Carlo (VMC) methods, making them competitive with leading computational methods such as coupled cluster theories and projector quantum Monte Carlo methods at both equilibrium and non-equilibrium geometries.

Key Contributions and Methodology

1. Flexibility and Accuracy of the Ansatz: The FermiNet is developed to address the limitations of conventional quantum chemistry methods that rely heavily on basis sets. The absence of a basis set approximation provides a distinct advantage as the FermiNet employs deep neural network architectures to approximate the wavefunction. This approach allows for high accuracy without the need for external data or system-specific tuning.
2. Neural Network Architecture: The FermiNet utilizes a deep neural network that integrates permutation-equivariant functions to construct a compact representation of the many-electron wavefunction. This is particularly suitable for fermionic systems as it naturally incorporates the antisymmetry required by the Fermi-Dirac statistics. The architecture combines one-electron and two-electron streams, leveraging coordinates and distances as inputs to model molecular interactions effectively.
3. Canonical Performance Against Established Methods: The FermiNet shows significant improvement over traditional Slater-Jastrow and Slater-Jastrow-backflow methods. In computational tests, the FermiNet demonstrated accuracy beyond that of CCSD(T) for systems at stretched bond lengths, such as the dissociation of the nitrogen molecule. It achieved this by modeling the electronic correlation more effectively through its architecture.
4. Scalability: A single FermiNet model was successfully applied to a diverse set of molecules without system-specific modifications, confirming its general applicability across a spectrum of quantum chemistry problems. The FermiNet's computational complexity scales quartically with the size of the system, making it an attractive option for larger electronic structures that are beyond reach for exact methods like FCI.
5. Optimization Techniques: The paper extends the Kronecker Factorized Approximate Curvature (KFAC) method for optimization in neural network training, enabling efficient convergence even for the complex, high-dimensional space of molecular wavefunctions.
6. Application and Future Potential: Besides providing competitive ground state energies, the FermiNet framework offers an avenue for exploring atomic and molecular properties by computing wavefunction-based observables directly, which is often challenging in quantum chemistry.

Implications and Prospective Developments

The FermiNet exemplifies the potential of neural network frameworks in computational quantum chemistry, offering a robust alternative to traditional methods bound by the limitations of predefined basis sets. Its adaptability to larger systems and accuracy in challenging electronic configurations suggest that the exploration of deeper, possibly wider neural networks could further enhance performance.

Theoretical implications extend to potentially reducing the computational overhead in associated electron dynamics studies and excited-state properties by exploiting the capabilities of neural networks in smoothly interpolating complex landscapes. Practically, this research advances the prospect of using machine learning for direct, real-time optimization of wavefunctions in quantum devices.

Furthermore, this work opens pathways to investigate other neural network architectures tailored to specific quantum calculations, such as mapping time-dependent properties or multi-reference ground states, providing a fertile ground for collaborative research between artificial intelligence and quantum mechanics.

In conclusion, by innovatively designing and employing deep neural networks as potent wavefunction approximators, the paper pushes the frontiers of quantum chemistry, demonstrating a significant step towards highly accurate, scalable quantum simulations that could redefine landscape in electronic structure calculations.