Machine learning for structure-property relationships: Scalability and limitations

TL;DR

The paper addresses the challenge of scalable prediction of intensive properties and phase classification in many-body systems by exploiting locality through a block-based approach with block size $\ell$ and averaging over blocks to reach large systems. It introduces a symmetry-aware lattice descriptor based on group-theoretical irreducible representations under $D_4$ and $Z_2$, including reference IR coefficients, fed into a CNN+FFN to predict block quantities $\mathcal Q_{\alpha}$. Results on the 2D Ising model reveal a scaling relation between block size and correlation length: the prediction error scales as $\varepsilon \sim \ell^{-\theta} \Phi(\ell/\xi)$ with $\theta \approx 0.12$ and the critical window scales as $\Delta T \sim \ell^{-1/\nu}$ with $\nu = 1$. The framework provides a practical, transferable path to large-scale predictions while clarifying fundamental scalability limits set by the diverging correlation length, and it points to extensions to electronic properties and unsupervised methods.

Abstract

We present a scalable machine learning (ML) framework for predicting intensive properties and particularly classifying phases of many-body systems. Scalability and transferability are central to the unprecedented computational efficiency of ML methods. In general, linear-scaling computation can be achieved through the divide and conquer approach, and the locality of physical properties is key to partitioning the system into sub-domains that can be solved separately. Based on the locality assumption, ML model is developed for the prediction of intensive properties of a finite-size block. Predictions of large-scale systems can then be obtained by averaging results of the ML model from randomly sampled blocks of the system. We show that the applicability of this approach depends on whether the block-size of the ML model is greater than the characteristic length scale of the system. In particular, in the case of phase identification across a critical point, the accuracy of the ML prediction is limited by the diverging correlation length. The two-dimensional Ising model is used to demonstrate the proposed framework. We obtain an intriguing scaling relation between the prediction accuracy and the ratio of ML block size over the spin-spin correlation length. Implications for practical applications are also discussed.

Figures (7)

• Figure 1: Machine learning model for phase classification or prediction of other intensive properties of a two-dimensional Ising model. The ML model is composed of two central components: the descriptor and the neural network. The input of the ML model is a square block of Ising spins with a linear size $\ell$. The descriptor corresponds to a representation of this spin-block that is invariant with respect to symmetry operations of the $D_4$ point group of the square lattice. Essentially, the eight symmetry-related configurations are mapped to the same feature variables $\bm G = (G_1, G_2, G_3, \cdots)$, which are then fed to the input layer of the NN. The output node of the NN gives the predicted intensive property.
• Figure 2: A scalable ML approach relies on the partitioning of the system into finite-size blocks that can be solved individually. The presumably time-consuming calculation of some physical properties of the block of linear size $\ell$ is encoded in the ML model, which is implemented using the multi-layer NN here. Panel (a) shows a schematic of the training process. The loss function $\mathcal{L}$ quantifies the difference between the exact value and the average of the ML predictions from each block. For application of the ML framework to large systems, one can design the partitioning such that the blocks cover the whole system. However, for extremely large system $L \gg \ell$, or experimental data where the system approaches the thermodynamic limit, a practical approach is to randomly select a large number of blocks to represent the system, as shown in panel (b). The estimation of the physical property is again the average of ML prediction from all blocks.
• Figure 3: Left panels show the energy density $\rho_E$ of the 2D Ising model as a function of temperature obtained from Monte Carlo (MC) simulations and ML predictions with block size (a) $\ell = 32$ and (b) $\ell = 64$. The right panels show the comparison of energy density predicted by ML models versus that obtained from MC simulations again for two different block sizes (c) $\ell = 32$ and (d) $\ell = 64$.
• Figure 4: Comparison of the block size $\ell$ and the correlation length $\xi$ of the 2D Ising model at (a) $T = 4.2J$, which is representative of the high-temperature paramagnetic phase, and (b) $T = 2.28J$ close to the critical point $T_c \approx 2.269J$.
• Figure 5: (a) Accuracy $\mathcal{A}$ of ML phase classification versus temperature for ML models of different block sizes $\ell$. (b) The temperature window $\Delta T = T^*_+ - T^*_-$ where the classification accuracy $\mathcal{A}$ drops below $85\%$ and $90\%$ versus the block size $\ell$. The dashed lines indicate the power-law behavior $\Delta T \sim \ell^{-1}$.