Analytic Continuation by Feature Learning

TL;DR

This work proposes a novel neural network architecture, named the Feature Learning Network (FL-net), to enhance the prediction accuracy of spectral functions, achieving an improvement of at least $20\% over traditional methods, such as the Maximum Entropy Method (MEM), and previous neural network approaches.

Abstract

Analytic continuation aims to reconstruct real-time spectral functions from imaginary-time Green's functions; however, this process is notoriously ill-posed and challenging to solve. We propose a novel neural network architecture, named the Feature Learning Network (FL-net), to enhance the prediction accuracy of spectral functions, achieving an improvement of at least $20\%$ over traditional methods, such as the Maximum Entropy Method (MEM), and previous neural network approaches. Furthermore, we develop an analytical method to evaluate the robustness of the proposed network. Using this method, we demonstrate that increasing the hidden dimensionality of FL-net, while leading to lower loss, results in decreased robustness. Overall, our model provides valuable insights into effectively addressing the complex challenges associated with analytic continuation.

Figures (9)

• Figure 1: Reconstruction of single-peak spectra: Left—the spectral function is reconstructed by learning its spectral features through feature learning; Right—our neural network reconstructs the spectral function by learning parameterized hidden variables $h$.
• Figure 2: FL-net: The network structure consists of two encoders (one for the imaginary-time Green's function and one for the spectral function) and a decoder. Both the encoder and decoder can have arbitrary architectures. The results consistently demonstrate that the loss from this approach is lower compared to direct learning methods.
• Figure 3: The red curve corresponds to the rapid decrease in loss, demonstrating effective learning with the following hidden dimensions: 8, 16, 23 for 8 peak samples; 6, 12, 17 for 6 peak samples; and 4, 8, 11 for 4 peak samples. The yellow curve represents nodes set to 64, 128, and 256, showing a comparatively slower decline in loss. The green curve, with remaining nodes set to 512 and 1024, shows an increase in loss, suggesting ineffective convergence or overfitting.
• Figure 4: (a) Relative loss comparison across datasets with varying numbers of peaks for different networks: fully connected network (FC net, blue), Fournier network (orange), direct network (D-net, green), and feature learning network (FL-net, red). FL-net achieves the lowest loss, reducing loss by approximately 20$\%$ compared to the Fournier network. (b) Prediction comparison between FL-net (red), Fournier network (blue), and the maximum entropy method (MEM, green) against the ground truth (gray). MEM and the Fournier network struggle with capturing multiple peaks and sharp spectral features, while FL-net consistently matches the true spectra, showing better accuracy in complex cases.
• Figure 5: (a) Prediction and singular value: The upper plot shows the true spectral results in gray and the predicted results as a blue dashed line. The lower plot has the horizontal axis representing the matrix column index and the vertical axis representing the corresponding singular values. Most of the column vectors $\boldsymbol{u}_j$ have singular values close to zero, indicating their minimal contribution. (b) The vertical axis shows the mode value and the horizontal axis indicates the mode range. The first six maximum singular values correspond to the left singular vector patterns: Different colors represent different modes, with significant fluctuations in each mode aligning well with features in the predicted results. (c) A scatter plot of the vertical axis $\tau_j$ versus horizontal axis $D_j$ association: Each point represents a mode.