Neural network backflow for ab initio quantum chemistry (2024)

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  Figure 1

  Illustration of the neural network backflow (NNBF) architecture with an example input configuration: two spin-orbitals with one spin-up electron occupying the first spin-orbital and one spin-down electron occupying the second spin-orbital. The neural network takes the configuration string as input and outputs a set of $D$ configuration-dependent single-particle orbitals of shape $(N_o,N_e)$. Rows of these orbitals are then selected based on the electron locations to form square matrices, from which determinants are computed. For this example, the first and last rows (the gray orbitals) are selected to compute the determinant. The sum of these determinants yields the amplitude for the input configuration.

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  Figure 2

  A diagrammatic description of the workflow of fixed-size selected configuration (FSSC). Each circle represents one configuration. Initially, the algorithm initializes a parameterized wave-function and a core space ${\mathcal{V}}^0$ of size $N_u=10$. After each iteration, the amplitude moduli for all configurations in ${\mathcal{V}}^{n-1}\cup {\mathcal{C}}^{n-1}$ are computed, and the 10 largest unique ones (denoted by red configurations) are selected to form the new core space ${\mathcal{V}}^n$. The energy (depicted as the loss function in the orange box) and its gradient are estimated by constraining the relative sum to only consider ${\mathcal{V}}^n$, and the latter is used to update the model parameters.

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  Figure 3

  Comparison between fixed-size selected configuration (FSSC) and Markov chain Monte Carlo (MCMC) schemes on lithium oxide ($N_o=30,N_e=14$, and $N_u=N_w=8192$). The red band represents the posttraining MCMC inference energy for the FSSC scheme with a width of $\sigma$ in each direction around the mean. A moving average window of 100 is applied to improve readability.

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  Figure 4

  Dissociation curve for ${\mathrm{N}}_2$ obtained with neural network backflow (NNBF), Hartree-Fock (HF), Coupled Cluster with Single and Double excitations (CCSD), and Coupled Cluster with Single, Double and perturbative Triple excitations (CCSD(T)) methods. The full configuration interaction (FCI) energy is used as the ground-truth energy. The NNBF energy is trained using the fixed-size selected configuration (FSSC) scheme with $N_u=4096<N_t$, and the reported energy here is computed exactly, as it remains feasible to compute.

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  Figure 5

  Effects of network architecture on the neural network backflow (NNBF) performance on ${\mathrm{CH}}_4$ ($N_o=18,N_e=10,N_u=N_t$, and $L=1$). Each point is one run of the same model. Double precision is employed for calculating the exact inference energy, as higher precision is necessary when the NNBF state closely approaches the true ground state. (a)Effect of network depth. The increase in performance levels off after the addition of two hidden layers to NNBF states. (b)Effect of number of hidden units. A wider hidden layer consistently improves accuracy, with the energy error decreasing at a rate of $O\left(h^{-2.256228}\right)$ with $R^2=0.968270$ until $h=64$. (c)Effect of number of determinants ($h=30$). Expanding the number of determinants reduces the energy error and begins to plateau after four determinants.

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  Figure 6

  Effects of network architecture on neural network backflow (NNBF) performance on the ${\mathrm{Li}}_2\mathrm{O}$ with $N_u=8192$. (a)Effect of network depth. The improvement with more layers quickly saturates after two layers are added to NNBF states. Baselines from other network quantum states (NNQS) works have been provided for comparison: QiankunNet [31], NAQS [28], MADE [29], ${\mathrm{QiankunNet}}^{*}$ [32], and RBM-SC [30]. The dark orange star denotes the best NNBF energy we have obtained with a greater $N_u$. (b)Effect of number of hidden units. Wider hidden layer continuously improves the accuracy with the absolute error drops at a rate of $O\left(h^{-0.671855}\right)$ with $R^2=0.952529$ until $h=256$. (c)Effect of number of determinants ($h=64$). Increasing the number of determinants reduces the energy error, but the improvement begins to plateau after reaching four determinants.

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  Figure 7

  (a)Experiments exploring the impact of batch size $N_u$ on energy improvements with $(L,h,D)=(2,64,1)$ and (2,256,1) on lithium oxide. The energy error decreases approximately following $O\left(N_u^{-1.137885}\right)$ with an $R^2$ value of 0.900605 for $h=64$ and $O\left(N_u^{-1.288627}\right)$ with $R^2=0.964273$ for $h=256$. Some data points from Fig.6are included for comparison in the inset. The black star represents a trial trained with $(L,h,D)=(2,64,1)$ using the sampling scheme from Ref.[30]. Its $x$ position is determined by the average batch size at convergence. The energy closely aligns with the fitted line for the fixed-size selected configuration (FSSC) scheme, indicating that using a dynamic batch size does not offer a noticeable improvement in training. (b)Demonstration of the effectiveness of the batch size scheduling (BSS) strategy, with a moving average window of 100 applied for improved readability. (c)Experiments examining the dependence of energy improvements on batch size $N_u$ with various $h$ values on methane. Each data point represents the average and the standard deviation of the exact neural network backflow (NNBF) energy across three seeds.

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  Figure 8

  Optimization progress for ${\mathrm{CH}}_4$, with and without configuration interaction with single and double excitations (CISD) pretraining ($L=1,h=512$, and $N_u=N_t$). The training iteration comprises 20000 steps, and a moving average window of 1000 is applied for better readability.

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