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arXiv:2205.08367 · Physical review. E

Deep-learning density functionals for gradient descent optimization

Emanuele Costa, Giuseppe Scriva, Rosario Fazio, Sebastiano Pilati

Published Oct 21, 2022• 5 citations
arXivHTMLPDF

Abstract

Machine-learned regression models represent a promising tool to implement accurate and computationally affordable energy-density functionals to solve quantum many-body problems via density functional theory. However, while they can easily be trained to accurately map ground-state density profiles to the corresponding energies, their functional derivatives often turn out to be too noisy, leading to instabilities in self-consistent iterations and in gradient-based searches of the ground-state density profile. We investigate how these instabilities occur when standard deep neural networks are adopted as regression models, and we show how to avoid them by using an ad hoc convolutional architecture featuring an interchannel averaging layer. The main testbed we consider is a realistic model for noninteracting atoms in optical speckle disorder. With the interchannel average, accurate and systematically improvable ground-state energies and density profiles are obtained via gradient-descent optimization, without instabilities nor violations of the variational principle.

DOI
10.1103/physreve.106.045309

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Machine-learned regression models represent a promising tool to implement accurate and computationally affordable energy-density functionals to solve quantum many-body problems via density functional theory. However, while they can easily be trained to accurately map ground-state density profiles to the corresponding energies, their functional derivatives often turn out to be too noisy, leading to instabilities in self-consistent iterations and in gradient-based searches of the ground-state density profile. We investigate how these instabilities occur when standard deep neural networks are adopted as regression models, and we show how to avoid them by using an ad hoc convolutional architecture featuring an interchannel averaging layer. The main testbed we consider is a realistic model for noninteracting atoms in optical speckle disorder. With the interchannel average, accurate and systematically improvable ground-state energies and density profiles are obtained via gradient-descent optimization, without instabilities nor violations of the variational principle.
Density functional theory (DFT) is routinely employed in material science and quantum chemistry to simulate weakly correlated electronic systems. Recently, deep learning (DL) techniques have been adopted to develop promising functionals for the strongly correlated regime. DFT can be applied to quantum spin models too, but functionals based on DL have not been developed yet. Here, we investigate DL-based DFTs for random quantum Ising chains, both with nearest-neighbor and up to next-nearest-neighbor couplings. Our neural functionals are trained and tested on data produced via the Jordan-Wigner transformation, exact diagonalization, and tensor-network methods. An economical gradient-descent optimization is used to find the ground-state properties of previously unseen Hamiltonian instances. Notably, our nonlocal functionals drastically improve upon the common local density approximations, and they are designed to be scalable, allowing us to simulate chain sizes much larger than those used for training. The prediction accuracy is analyzed, paying attention to the spatial correlations of the learned functionals and to the role of quantum criticality. Our findings indicate a suitable strategy to extend the reach of other computational methods with a controllable accuracy.
The solution of complex many-body lattice models can often be found by defining an energy functional of the relevant density of the problem. For instance, in the case of the Hubbard model the spin-resolved site occupation is enough to describe the system's total energy. Similarly to standard density functional theory, however, the exact functional is unknown, and suitable approximations need to be formulated. By using a deep-learning neural network trained on exact-diagonalization results, we demonstrate that one can construct an exact functional for the Hubbard model. In particular, we show that the neural network returns a ground-state energy numerically indistinguishable from that obtained by exact diagonalization and, most importantly, that the functional satisfies the two Hohenberg-Kohn theorems: for a given ground-state density it yields the external potential, and it is fully variational in the site occupation.
We present a novel approach that combines machine learning with direct variational energy optimization via the density matrix to solve the Kohn-Sham equation in density functional theory. Instead of relying on the conventional self-consistent field method, our approach directly optimizes the ground state by predicting the density matrix using a neural network, thereby bypassing Hamiltonian matrix diagonalization. Our model employs equivariant neural networks to generate a physically constrained density matrix, enabling stable and efficient energy minimization. This method integrates the construction of training sets directly into the model training process and achieves high accuracy in predicting ground-state properties across various molecular and extended systems, thereby establishing a powerful machine learning paradigm for electronic structure optimization and paving the way for large-scale quantum simulations.
Supervised machine learning is emerging as a powerful computational tool to predict the properties of complex quantum systems at a limited computational cost. In this article, we quantify how accurately deep neural networks can learn the properties of disordered quantum systems as a function of the system size. We implement a scalable convolutional network that can address arbitrary system sizes. This network is compared with a recently introduced extensive convolutional architecture [Mills et al., Chem. Sci. 10, 4129 (2019)2041-652010.1039/C8SC04578J] and with conventional dense networks with all-to-all connectivity. The networks are trained to predict the exact ground-state energies of various disordered systems, namely, a continuous-space single-particle Hamiltonian for cold-atoms in speckle disorder, and different setups of a quantum Ising chain with random couplings, including one with only short-range interactions and one augmented with a long-range term. In all testbeds we consider, the scalable network retains high accuracy as the system size increases. Furthermore, we demonstrate that the network scalability enables a transfer-learning protocol, whereby a pretraining performed on small systems drastically accelerates the learning of large-system properties, allowing reaching high accuracy with small training sets. In fact, with the scalable network one can even extrapolate to sizes larger than those included in the training set, accurately reproducing the results of state-of-the-art quantum Monte Carlo simulations.
Density estimation is a fundamental task in statistics and machine learning that aims to estimate, from a set of samples, the probability density function of the distribution that generated them. There are different methods for addressing this problem but recently deep-neural density estimation methods have emerged as a powerful alternative. This paper presents a novel method for neural density estimation based on density matrices and adaptive Fourier features. Density matrices are commonly used in quantum mechanics to represent the quantum state of a physical system. In this work, they are used to estimate probability densities using an operation called quantum measurement. The proposed method can be trained without optimization using an averaging operation over the samples of the training dataset. It can also be integrated with deep learning architectures and trained using gradient descent. The performance of the proposed method was evaluated on a range of synthetic and real datasets and compared with fast kernel density estimation and state-of-the-art neural density estimation methods. The results demonstrate that the proposed method achieves competitive performance while being faster and more efficient than existing methods.
Abstract Deep neural networks (DNNs) and auto differentiation have been widely used in computational physics to solve variational problems. When a DNN is used to represent the wave function and solve quantum many-body problems using variational optimization, various physical constraints have to be injected into the neural network by construction to increase the data and learning efficiency. We build the unitary constraint to the variational wave function using a monotonic neural network to represent the cumulative distribution function (CDF) . Using this constrained neural network to represent the variational wave function, we solve Schrodinger equations using auto-differentiation and stochastic gradient descent (SGD) by minimizing the violation of the trial wave function to the Schrodinger equation. For several classical problems in quantum mechanics, we obtain their ground state wave function and energy with very low errors. The method developed in the present paper may pave a new way for solving nuclear many-body problems in the future.