The type of symmetry is typically fixed and has to be chosen in advance.

First, it requires efficient and flexible parameterisations of layer-wise equivari-ances.

Deep learning models have proven enormously successful at using multiple layers of representation to learn relevant features of structured data. Encoding physical symmetries into these models can improve performance on difficult tasks, and recent work has motivated the principle of parameter symmetry breaking and restoration as a unifying mechanism underlying their hierarchical learning dynamics. We evaluate the role of parameter symmetry and network expressivity in the generalisation behaviour of neural networks when learning a real-space renormalisation group (RG) transformation, using the central limit theorem (CLT) as a test case map. We consider simple multilayer perceptrons (MLPs) and graph neural networks (GNNs), and vary weight symmetries and activation functions across architectures. Our results reveal a competition between symmetry constraints and expressivity, with overly complex or overconstrained models generalising poorly. We analytically demonstrate this poor generalisation behaviour for certain constrained MLP architectures by recasting the CLT as a cumulant recursion relation and making use of an established framework to propagate cumulants through MLPs. We also empirically validate an extension of this framework from MLPs to GNNs, elucidating the internal information processing performed by these more complex models. These findings offer new insight into the learning dynamics of symmetric networks and their limitations in modelling structured physical transformations.

Simple decision heuristics are models of human and animal behavior that use few pieces of information--perhaps only a single piece of information--and integrate the pieces in simple ways, for example, by considering them sequentially, one at a time, or by giving them equal weight. We focus on three families of heuristics: single-cue decision making, lexicographic decision making, and tallying. It is unknown how quickly these heuristics can be learned from experience. We show, analytically and empirically, that substantial progress in learning can be made with just a few training samples. When training samples are very few, tallying performs substantially better than the alternative methods tested. Our empirical analysis is the most extensive to date, employing 63 natural data sets on diverse subjects.

Generative model for 2D materials has shown significant promise in accelerating the material discovery process. The stability and performance of these materials are strongly influenced by their underlying symmetry. However, existing generative models for 2D materials often neglect symmetry constraints, which limits both the diversity and quality of the generated structures. Here, we introduce a symmetry-constrained diffusion model (SCDM) that integrates space group symmetry into the generative process. By incorporating Wyckoff positions, the model ensures adherence to symmetry principles, leading to the generation of 2,000 candidate structures. DFT calculations were conducted to evaluate the convex hull energies of these structures after structural relaxation. From the generated samples, 843 materials that met the energy stability criteria (Ehull < 0.6 eV/atom) were identified. Among these, six candidates were selected for further stability analysis, including phonon band structure evaluations and electronic properties investigations, all of which exhibited phonon spectrum stability. To benchmark the performance of SCDM, a symmetry-unconstrained diffusion model was also evaluated via crystal structure prediction model. The results highlight that incorporating symmetry constraints enhances the effectiveness of generated 2D materials, making a contribution to the discovery of 2D materials through generative modeling.

Simple decision heuristics are models of human and animal behavior that use few pieces of information--perhaps only a single piece of information--and integrate the pieces in simple ways, for example, by considering them sequentially, one at a time, or by giving them equal weight. We focus on three families of heuristics: single-cue decision making, lexicographic decision making, and tallying. It is unknown how quickly these heuristics can be learned from experience. We show, analytically and empirically, that substantial progress in learning can be made with just a few training samples. When training samples are very few, tallying performs substantially better than the alternative methods tested. Our empirical analysis is the most extensive to date, employing 63 natural data sets on diverse subjects.

In recent years, analog circuits have received extensive attention and are widely used in many emerging applications. The high demand for analog circuits necessitates shorter circuit design cycles. To achieve the desired performance and specifications, various geometrical symmetry constraints must be carefully considered during the analog layout process. However, the manual labeling of these constraints by experienced analog engineers is a laborious and time-consuming process. To handle the costly runtime issue, we propose a graph-based learning framework to automatically extract symmetric constraints in analog circuit layout. The proposed framework leverages the connection characteristics of circuits and the devices'information to learn the general rules of symmetric constraints, which effectively facilitates the extraction of device-level constraints on circuit netlists. The experimental results demonstrate that compared to state-of-the-art symmetric constraint detection approaches, our framework achieves higher accuracy and lower false positive rate.

The discovery of new functional and stable materials is a big challenge due to its complexity. This work aims at the generation of new crystal structures with desired properties, such as chemical stability and specified chemical composition, by using machine learning generative models. Compared to the generation of molecules, crystal structures pose new difficulties arising from the periodic nature of the crystal and from the specific symmetry constraints related to the space group. In this work, score-based probabilistic models based on annealed Langevin dynamics, which have shown excellent performance in various applications, are adapted to the task of crystal generation. The novelty of the presented approach resides in the fact that the lattice of the crystal cell is not fixed. During the training of the model, the lattice is learned from the available data, whereas during the sampling of a new chemical structure, two denoising processes are used in parallel to generate the lattice along the generation of the atomic positions. A multigraph crystal representation is introduced that respects symmetry constraints, yielding computational advantages and a better quality of the sampled structures. We show that our model is capable of generating new candidate structures in any chosen chemical system and crystal group without any additional training. To illustrate the functionality of the proposed method, a comparison of our model to other recent generative models, based on descriptor-based metrics, is provided.