Search

As the dielectric film thickness shrinks to ~10 nm, some traditional wurtzite piezoelectric materials demonstrate ferroelectricity through element doping. Among them, Sc doped AlN and Mg doped ZnO are the most famous examples. While it is widely acknowledged that the dopant atoms effectively reduce the coercive field, enabling ferroelectric polarization switching, the material genome of these wurtzite (WZ) ferroelectrics is still less understood. In this work, we analyze the features of WZ ferroelectrics, ascribing them to five-coordination (5C) ferroelectrics, which may be compared with 6C ferroelectrics (perovskite-type) and 7C ferroelectrics (hafnia-like). In particular, the exact reason for their adopting the hexagonal WZ structure instead of the zinc blende structure is studied. Emphasis is paid to the degree of ionicity in promoting the hexagonal arrangement, and the phenomenon of layer distance compression is discovered and explained in WZ ferroelectrics. The role of element doping in coercive field reduction is understood within this context., Comment: 5 figures

A spike camera is a specialized high-speed visual sensor that offers advantages such as high temporal resolution and high dynamic range compared to conventional frame cameras. These features provide the camera with significant advantages in many computer vision tasks. However, the tasks of novel view synthesis based on spike cameras remain underdeveloped. Although there are existing methods for learning neural radiance fields from spike stream, they either lack robustness in extremely noisy, low-quality lighting conditions or suffer from high computational complexity due to the deep fully connected neural networks and ray marching rendering strategies used in neural radiance fields, making it difficult to recover fine texture details. In contrast, the latest advancements in 3DGS have achieved high-quality real-time rendering by optimizing the point cloud representation into Gaussian ellipsoids. Building on this, we introduce SpikeGS, the method to learn 3D Gaussian fields solely from spike stream. We designed a differentiable spike stream rendering framework based on 3DGS, incorporating noise embedding and spiking neurons. By leveraging the multi-view consistency of 3DGS and the tile-based multi-threaded parallel rendering mechanism, we achieved high-quality real-time rendering results. Additionally, we introduced a spike rendering loss function that generalizes under varying illumination conditions. Our method can reconstruct view synthesis results with fine texture details from a continuous spike stream captured by a moving spike camera, while demonstrating high robustness in extremely noisy low-light scenarios. Experimental results on both real and synthetic datasets demonstrate that our method surpasses existing approaches in terms of rendering quality and speed. Our code will be available at https://github.com/520jz/SpikeGS., Comment: Accepted by ACCV 2024

For image inpainting, the existing Denoising Diffusion Probabilistic Model (DDPM) based method i.e. RePaint can produce high-quality images for any inpainting form. It utilizes a pre-trained DDPM as a prior and generates inpainting results by conditioning on the reverse diffusion process, namely denoising process. However, this process is significantly time-consuming. In this paper, we propose an efficient DDPM-based image inpainting method which includes three speed-up strategies. First, we utilize a pre-trained Light-Weight Diffusion Model (LWDM) to reduce the number of parameters. Second, we introduce a skip-step sampling scheme of Denoising Diffusion Implicit Models (DDIM) for the denoising process. Finally, we propose Coarse-to-Fine Sampling (CFS), which speeds up inference by reducing image resolution in the coarse stage and decreasing denoising timesteps in the refinement stage. We conduct extensive experiments on both faces and general-purpose image inpainting tasks, and our method achieves competitive performance with approximately 60 times speedup., Comment: The code is avaliable at: https://github.com/linghuyuhangyuan/M2S

When dealing with the task of fine-grained scene image classification, most previous works lay much emphasis on global visual features when doing multi-modal feature fusion. In other words, models are deliberately designed based on prior intuitions about the importance of different modalities. In this paper, we present a new multi-modal feature fusion approach named MAA (Modality-Agnostic Adapter), trying to make the model learn the importance of different modalities in different cases adaptively, without giving a prior setting in the model architecture. More specifically, we eliminate the modal differences in distribution and then use a modality-agnostic Transformer encoder for a semantic-level feature fusion. Our experiments demonstrate that MAA achieves state-of-the-art results on benchmarks by applying the same modalities with previous methods. Besides, it is worth mentioning that new modalities can be easily added when using MAA and further boost the performance. Code is available at https://github.com/quniLcs/MAA.

This paper presents a comprehensive exploration of relation extraction utilizing advanced language models, specifically Chain of Thought (CoT) and Graphical Reasoning (GRE) techniques. We demonstrate how leveraging in-context learning with GPT-3.5 can significantly enhance the extraction process, particularly through detailed example-based reasoning. Additionally, we introduce a novel graphical reasoning approach that dissects relation extraction into sequential sub-tasks, improving precision and adaptability in processing complex relational data. Our experiments, conducted on multiple datasets, including manually annotated data, show considerable improvements in performance metrics, underscoring the effectiveness of our methodologies.

The conformational landscape of proteins is crucial to understanding their functionality in complex biological processes. Traditional physics-based computational methods, such as molecular dynamics (MD) simulations, suffer from rare event sampling and long equilibration time problems, hindering their applications in general protein systems. Recently, deep generative modeling techniques, especially diffusion models, have been employed to generate novel protein conformations. However, existing score-based diffusion methods cannot properly incorporate important physical prior knowledge to guide the generation process, causing large deviations in the sampled protein conformations from the equilibrium distribution. In this paper, to overcome these limitations, we propose a force-guided SE(3) diffusion model, ConfDiff, for protein conformation generation. By incorporating a force-guided network with a mixture of data-based score models, ConfDiff can generate protein conformations with rich diversity while preserving high fidelity. Experiments on a variety of protein conformation prediction tasks, including 12 fast-folding proteins and the Bovine Pancreatic Trypsin Inhibitor (BPTI), demonstrate that our method surpasses the state-of-the-art method., Comment: ICML 2024

Visual Place Recognition (VPR) is crucial in computer vision, aiming to retrieve database images similar to a query image from an extensive collection of known images. However, like many vision tasks, VPR always degrades at night due to the scarcity of nighttime images. Moreover, VPR needs to address the cross-domain problem of night-to-day rather than just the issue of a single nighttime domain. In response to these issues, we present NocPlace, which leverages generative and inherited knowledge transfer to embed resilience against dazzling lights and extreme darkness in the global descriptor. First, we establish a day-night urban scene dataset called NightCities, capturing diverse lighting variations and dark scenarios across 60 cities globally. Then, an image generation network is trained on this dataset and processes a large-scale VPR dataset, obtaining its nighttime version. Finally, VPR models are fine-tuned using descriptors inherited from themselves and night-style images, which builds explicit cross-domain contrastive relationships. Comprehensive experiments on various datasets demonstrate our contributions and the superiority of NocPlace. Without adding any real-time computing resources, NocPlace improves the performance of Eigenplaces by 7.6% on Tokyo 24/7 Night and 16.8% on SVOX Night., Comment: 28 pages,9 figures

Driven by abundant satellite imagery, machine learning-based approaches have recently been promoted to generate high-resolution crop cultivation maps to support many agricultural applications. One of the major challenges faced by these approaches is the limited availability of ground truth labels. In the absence of ground truth, existing work usually adopts the "direct transfer strategy" that trains a classifier using historical labels collected from other regions and then applies the trained model to the target region. Unfortunately, the spectral features of crops exhibit inter-region and inter-annual variability due to changes in soil composition, climate conditions, and crop progress, the resultant models perform poorly on new and unseen regions or years. Despite recent efforts, such as the application of the deep adaptation neural network (DANN) model structure in the deep adaptation crop classification network (DACCN), to tackle the above cross-domain challenges, their effectiveness diminishes significantly when there is a large dissimilarity between the source and target regions. This paper introduces the Crop Mapping Spectral-temporal Generative Adversarial Neural Network (CropSTGAN), a novel solution for cross-domain challenges, that doesn't require target domain labels. CropSTGAN learns to transform the target domain's spectral features to those of the source domain, effectively bridging large dissimilarities. Additionally, it employs an identity loss to maintain the intrinsic local structure of the data. Comprehensive experiments across various regions and years demonstrate the benefits and effectiveness of the proposed approach. In experiments, CropSTGAN is benchmarked against various state-of-the-art (SOTA) methods. Notably, CropSTGAN significantly outperforms these methods in scenarios with large data distribution dissimilarities between the target and source domains.

We present a novel method for 3D surface reconstruction from multiple images where only a part of the object of interest is captured. Our approach builds on two recent developments: surface reconstruction using neural radiance fields for the reconstruction of the visible parts of the surface, and guidance of pre-trained 2D diffusion models in the form of Score Distillation Sampling (SDS) to complete the shape in unobserved regions in a plausible manner. We introduce three components. First, we suggest employing normal maps as a pure geometric representation for SDS instead of color renderings which are entangled with the appearance information. Second, we introduce the freezing of the SDS noise during training which results in more coherent gradients and better convergence. Third, we propose Multi-View SDS as a way to condition the generation of the non-observable part of the surface without fine-tuning or making changes to the underlying 2D Stable Diffusion model. We evaluate our approach on the BlendedMVS dataset demonstrating significant qualitative and quantitative improvements over competing methods.

We develop a method that recovers the surface, materials, and illumination of a scene from its posed multi-view images. In contrast to prior work, it does not require any additional data and can handle glossy objects or bright lighting. It is a progressive inverse rendering approach, which consists of three stages. First, we reconstruct the scene radiance and signed distance function (SDF) with our novel regularization strategy for specular reflections. Our approach considers both the diffuse and specular colors, which allows for handling complex view-dependent lighting effects for surface reconstruction. Second, we distill light visibility and indirect illumination from the learned SDF and radiance field using learnable mapping functions. Third, we design a method for estimating the ratio of incoming direct light represented via Spherical Gaussians reflected in a specular manner and then reconstruct the materials and direct illumination of the scene. Experimental results demonstrate that the proposed method outperforms the current state-of-the-art in recovering surfaces, materials, and lighting without relying on any additional data., Comment: 12 pages, 10 figures. Project page: https://authors-hub.github.io/Factored-NeuS

A signed distance function (SDF) parametrized by an MLP is a common ingredient of neural surface reconstruction. We build on the successful recent method NeuS to extend it by three new components. The first component is to borrow the tri-plane representation from EG3D and represent signed distance fields as a mixture of tri-planes and MLPs instead of representing it with MLPs only. Using tri-planes leads to a more expressive data structure but will also introduce noise in the reconstructed surface. The second component is to use a new type of positional encoding with learnable weights to combat noise in the reconstruction process. We divide the features in the tri-plane into multiple frequency scales and modulate them with sin and cos functions of different frequencies. The third component is to use learnable convolution operations on the tri-plane features using self-attention convolution to produce features with different frequency bands. The experiments show that PET-NeuS achieves high-fidelity surface reconstruction on standard datasets. Following previous work and using the Chamfer metric as the most important way to measure surface reconstruction quality, we are able to improve upon the NeuS baseline by 57% on Nerf-synthetic (0.84 compared to 1.97) and by 15.5% on DTU (0.71 compared to 0.84). The qualitative evaluation reveals how our method can better control the interference of high-frequency noise. Code available at \url{https://github.com/yiqun-wang/PET-NeuS}., Comment: CVPR 2023; 20 Pages; Project page: \url{https://github.com/yiqun-wang/PET-NeuS}

Learning-based point cloud registration methods can handle clean point clouds well, while it is still challenging to generalize to noisy, partial, and density-varying point clouds. To this end, we propose a novel point cloud registration framework for these imperfect point clouds. By introducing a neural implicit representation, we replace the problem of rigid registration between point clouds with a registration problem between the point cloud and the neural implicit function. We then propose to alternately optimize the implicit function and the registration between the implicit function and point cloud. In this way, point cloud registration can be performed in a coarse-to-fine manner. By fully capitalizing on the capabilities of the neural implicit function without computing point correspondences, our method showcases remarkable robustness in the face of challenges such as noise, incompleteness, and density changes of point clouds.

Molecular representation learning plays a crucial role in AI-assisted drug discovery research. Encoding 3D molecular structures through Euclidean neural networks has become the prevailing method in the geometric deep learning community. However, the equivariance constraints and message passing in Euclidean space may limit the network expressive power. In this work, we propose a Harmonic Molecular Representation learning (HMR) framework, which represents a molecule using the Laplace-Beltrami eigenfunctions of its molecular surface. HMR offers a multi-resolution representation of molecular geometric and chemical features on 2D Riemannian manifold. We also introduce a harmonic message passing method to realize efficient spectral message passing over the surface manifold for better molecular encoding. Our proposed method shows comparable predictive power to current models in small molecule property prediction, and outperforms the state-of-the-art deep learning models for ligand-binding protein pocket classification and the rigid protein docking challenge, demonstrating its versatility in molecular representation learning., Comment: 25 pages including Appendix

3D-aware image synthesis has attracted increasing interest as it models the 3D nature of our real world. However, performing realistic object-level editing of the generated images in the multi-object scenario still remains a challenge. Recently, a 2D GAN termed BlobGAN has demonstrated great multi-object editing capabilities on real-world indoor scene datasets. In this work, we propose BlobGAN-3D, which is a 3D-aware improvement of the original 2D BlobGAN. We enable explicit camera pose control while maintaining the disentanglement for individual objects in the scene by extending the 2D blobs into 3D blobs. We keep the object-level editing capabilities of BlobGAN and in addition allow flexible control over the 3D location of the objects in the scene. We test our method on real-world indoor datasets and show that our method can achieve comparable image quality compared to the 2D BlobGAN and other 3D-aware GAN baselines while being able to enable camera pose control and object-level editing in the challenging multi-object real-world scenarios.

Partial-label learning (PLL) is an important weakly supervised learning problem, which allows each training example to have a candidate label set instead of a single ground-truth label. Identification-based methods have been widely explored to tackle label ambiguity issues in PLL, which regard the true label as a latent variable to be identified. However, identifying the true labels accurately and completely remains challenging, causing noise in pseudo labels during model training. In this paper, we propose a new method called CroSel, which leverages historical predictions from the model to identify true labels for most training examples. First, we introduce a cross selection strategy, which enables two deep models to select true labels of partially labeled data for each other. Besides, we propose a novel consistency regularization term called co-mix to avoid sample waste and tiny noise caused by false selection. In this way, CroSel can pick out the true labels of most examples with high precision. Extensive experiments demonstrate the superiority of CroSel, which consistently outperforms previous state-of-the-art methods on benchmark datasets. Additionally, our method achieves over 90\% accuracy and quantity for selecting true labels on CIFAR-type datasets under various settings., Comment: Accepted by CVPR 2024

In recent years, supervised or unsupervised learning-based MVS methods achieved excellent performance compared with traditional methods. However, these methods only use the probability volume computed by cost volume regularization to predict reference depths and this manner cannot mine enough information from the probability volume. Furthermore, the unsupervised methods usually try to use two-step or additional inputs for training which make the procedure more complicated. In this paper, we propose the DS-MVSNet, an end-to-end unsupervised MVS structure with the source depths synthesis. To mine the information in probability volume, we creatively synthesize the source depths by splattering the probability volume and depth hypotheses to source views. Meanwhile, we propose the adaptive Gaussian sampling and improved adaptive bins sampling approach that improve the depths hypotheses accuracy. On the other hand, we utilize the source depths to render the reference images and propose depth consistency loss and depth smoothness loss. These can provide additional guidance according to photometric and geometric consistency in different views without additional inputs. Finally, we conduct a series of experiments on the DTU dataset and Tanks & Temples dataset that demonstrate the efficiency and robustness of our DS-MVSNet compared with the state-of-the-art methods., Comment: Accepted by ACM MM 2022

A very recent trend in generative modeling is building 3D-aware generators from 2D image collections. To induce the 3D bias, such models typically rely on volumetric rendering, which is expensive to employ at high resolutions. During the past months, there appeared more than 10 works that address this scaling issue by training a separate 2D decoder to upsample a low-resolution image (or a feature tensor) produced from a pure 3D generator. But this solution comes at a cost: not only does it break multi-view consistency (i.e. shape and texture change when the camera moves), but it also learns the geometry in a low fidelity. In this work, we show that it is possible to obtain a high-resolution 3D generator with SotA image quality by following a completely different route of simply training the model patch-wise. We revisit and improve this optimization scheme in two ways. First, we design a location- and scale-aware discriminator to work on patches of different proportions and spatial positions. Second, we modify the patch sampling strategy based on an annealed beta distribution to stabilize training and accelerate the convergence. The resulted model, named EpiGRAF, is an efficient, high-resolution, pure 3D generator, and we test it on four datasets (two introduced in this work) at $256^2$ and $512^2$ resolutions. It obtains state-of-the-art image quality, high-fidelity geometry and trains ${\approx} 2.5 \times$ faster than the upsampler-based counterparts. Project website: https://universome.github.io/epigraf., Comment: NeurIPS 2022

Neural rendering can be used to reconstruct implicit representations of shapes without 3D supervision. However, current neural surface reconstruction methods have difficulty learning high-frequency geometry details, so the reconstructed shapes are often over-smoothed. We develop HF-NeuS, a novel method to improve the quality of surface reconstruction in neural rendering. We follow recent work to model surfaces as signed distance functions (SDFs). First, we offer a derivation to analyze the relationship between the SDF, the volume density, the transparency function, and the weighting function used in the volume rendering equation and propose to model transparency as transformed SDF. Second, we observe that attempting to jointly encode high-frequency and low-frequency components in a single SDF leads to unstable optimization. We propose to decompose the SDF into a base function and a displacement function with a coarse-to-fine strategy to gradually increase the high-frequency details. Finally, we design an adaptive optimization strategy that makes the training process focus on improving those regions near the surface where the SDFs have artifacts. Our qualitative and quantitative results show that our method can reconstruct fine-grained surface details and obtain better surface reconstruction quality than the current state of the art. Code available at https://github.com/yiqun-wang/HFS., Comment: To appear in NeurIPS 2022. Project page: https://github.com/yiqun-wang/HFS

Limited by the computational efficiency and accuracy, generating complex 3D scenes remains a challenging problem for existing generation networks. In this work, we propose DepthGAN, a novel method of generating depth maps with only semantic layouts as input. First, we introduce a well-designed cascade of transformer blocks as our generator to capture the structural correlations in depth maps, which makes a balance between global feature aggregation and local attention. Meanwhile, we propose a cross-attention fusion module to guide edge preservation efficiently in depth generation, which exploits additional appearance supervision information. Finally, we conduct extensive experiments on the perspective views of the Structured3d panorama dataset and demonstrate that our DepthGAN achieves superior performance both on quantitative results and visual effects in the depth generation task.Furthermore, 3D indoor scenes can be reconstructed by our generated depth maps with reasonable structure and spatial coherency., Comment: 7 pages, 5 figures

Depth estimation is a crucial step for 3D reconstruction with panorama images in recent years. Panorama images maintain the complete spatial information but introduce distortion with equirectangular projection. In this paper, we propose an ACDNet based on the adaptively combined dilated convolution to predict the dense depth map for a monocular panoramic image. Specifically, we combine the convolution kernels with different dilations to extend the receptive field in the equirectangular projection. Meanwhile, we introduce an adaptive channel-wise fusion module to summarize the feature maps and get diverse attention areas in the receptive field along the channels. Due to the utilization of channel-wise attention in constructing the adaptive channel-wise fusion module, the network can capture and leverage the cross-channel contextual information efficiently. Finally, we conduct depth estimation experiments on three datasets (both virtual and real-world) and the experimental results demonstrate that our proposed ACDNet substantially outperforms the current state-of-the-art (SOTA) methods. Our codes and model parameters are accessed in https://github.com/zcq15/ACDNet., Comment: 13 pages, 6 figures

Materials property predictions have improved from advances in machine learning algorithms, delivering materials discoveries and novel insights through data-driven models of structure-property relationships. Nearly all available models rely on featurization of materials composition, however, whether the exclusive use of structural knowledge in such models has the capacity to make comparable predictions remains unknown. Here we employ a deep neural network model to decode structure-property relationships in crystalline materials without explicitly considering chemical compositions. The focus is on classification of crystal systems, mechanical elasticity, electronic band gap, and phase stability. Our model utilizes a three-dimensional (3D) momentum space representation of structure from elastic x-ray scattering theory that exhibits rotation and permutation invariance. We perform novel ablation studies to help interpret the model performance by perturbing the physically meaningful input features (i.e., the diffraction patterns) instead of tuning the architecture of the learning model as in conventional ablation methods. We find that the spatial symmetry of the 3D diffraction patterns, which reflects crystalline symmetry operations, is more important than the diffraction intensities contained within for the model to make a successful classification. Our work showcases the potential of using statistical learning models to help understand materials physics, rather than performing predictive and generative tasks as in most materials informatics research. We also argue that learning the crystal structure genome in a chemistry-agnostic manner demonstrates that some crystal structures inherently host high propensities for optimal materials properties, which enables the decoupling of structure and composition for future codesign of multifunctionality., Comment: 14 pages, 6 Figures

Electronic materials exhibiting phase transitions between metastable states (e.g., metal-insulator transition materials with abrupt electrical resistivity transformations) are challenging to decode. For these materials, conventional machine learning methods display limited predictive capability due to data scarcity and the absence of features impeding model training. In this article, we demonstrate a discovery strategy based on multi-objective Bayesian optimization to directly circumvent these bottlenecks by utilizing latent variable Gaussian processes combined with high-fidelity electronic structure calculations for validation in the chalcogenide lacunar spinel family. We directly and simultaneously learn phase stability and band gap tunability from chemical composition alone to efficiently discover all superior compositions on the design Pareto front. Previously unidentified electronic transitions also emerge from our featureless adaptive optimization engine. Our methodology readily generalizes to optimization of multiple properties, enabling co-design of complex multifunctional materials, especially where prior data is sparse.

We propose a novel framework for computing descriptors for characterizing points on three-dimensional surfaces. First, we present a new non-learned feature that uses graph wavelets to decompose the Dirichlet energy on a surface. We call this new feature wavelet energy decomposition signature (WEDS). Second, we propose a new multiscale graph convolutional network (MGCN) to transform a non-learned feature to a more discriminative descriptor. Our results show that the new descriptor WEDS is more discriminative than the current state-of-the-art non-learned descriptors and that the combination of WEDS and MGCN is better than the state-of-the-art learned descriptors. An important design criterion for our descriptor is the robustness to different surface discretizations including triangulations with varying numbers of vertices. Our results demonstrate that previous graph convolutional networks significantly overfit to a particular resolution or even a particular triangulation, but MGCN generalizes well to different surface discretizations. In addition, MGCN is compatible with previous descriptors and it can also be used to improve the performance of other descriptors, such as the heat kernel signature, the wave kernel signature, or the local point signature., Comment: Accepted to SIGGRAPH 2020. (15 pages, 15 figures, 12 tables, low-resolution version)

We perform systematic density functional theory (DFT) calculations to assess the performance of various exchange-correlation potentials $V_{xc}$ in describing the chalcogenide GaM$_4$Q$_8$ lacunar spinels (M=Mo, V, Nb, Ta; Q=S, Se). We examine the dependency of crystal structure (in cubic and rhombohedral symmetries), electronic structure, magnetism, optical conductivity, and lattice dynamics in lacunar spinels at four different levels of $V_{xc}$: the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA, and hybrid with fractional Fock exchange. We find that LDA significantly underperforms GGA and higher level functionals in predicting lattice constants. LDA also fails to properly describe the magnetism in the family, and for some compositions it does not find the ground state distorted crystal structure. The Perdew-Burke-Ernzerhof (PBE) and PBE revised for solids (PBEsol) GGA functionals perform reasonably well in predicting lattice constants as well as the electronic structures. We find that the GGA with an on-site Coulomb interaction (GGA$+U$) is unnecessary to produce a semiconducting state in the distorted polar $R3m$ phase. Plus Hubbard $U$ values ranging from 1$\sim$2 eV, however, improve the quantitative performance of the GGA functionals. The meta-GGA functional SCAN predicts reasonable lattice constants and electronic structures; it exhibits behavior similar to the GGA$+U$ functionals for small $U$ values. The hybrid functional HSE06 is accurate in predicting the lattice constants, but leads to a band gap of $\approx$1 eV in the rhombohedral phase, which is higher than the experimental estimation of 0.2 eV. Our findings suggest that accurate qualitative and quantitative simulations of the lacunar spinel family with DFT requires careful attention to the nuances of the exchange-correlation functional and considered spin structures., Comment: 16 Pages, 13 Figures, 5 Tables

We showcase the potential of symbolic regression as an analytic method for use in materials research. First, we briefly describe the current state-of-the-art method, genetic programming-based symbolic regression (GPSR), and recent advances in symbolic regression techniques. Next, we discuss industrial applications of symbolic regression and its potential applications in materials science. We then present two GPSR use-cases: formulating a transformation kinetics law and showing the learning scheme discovers the well-known Johnson-Mehl-Avrami-Kolmogorov (JMAK) form, and learning the Landau free energy functional form for the displacive tilt transition in perovskite LaNiO$_3$. Finally, we propose that symbolic regression techniques should be considered by materials scientists as an alternative to other machine-learning-based regression models for learning from data., Comment: 14 pages, 6 figures