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Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present a machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on graph neural networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. [ABSTRACT FROM AUTHOR]

Surface reconstruction from scattered point clouds is the process of generating surfaces from unstructured data configurations retrieved using an acquisition device such as a laser scanner. Smooth surfaces are possible with the use of spline representations, an established mathematical tool in computer-aided design and related application areas. One key step in the surface reconstruction process is the parameterization of the points, that is, the construction of a proper mapping of the 3D point cloud to a planar domain that preserves surface boundary and interior points. Despite achieving a remarkable progress, existing heuristics for generating a suitable parameterization face challenges related to the accuracy, the robustness with respect to noise, and the computational efficiency of the results. In this work, we propose a boundary-informed dynamic graph convolutional network (BIDGCN) characterized by a novel boundary-informed input layer, with special focus on applications related to adaptive spline approximation of scattered data. The newly introduced layer propagates given boundary information to the interior of the point cloud, in order to let the input data be suitably processed by successive graph convolutional network layers. We apply our BIDGCN model to the problem of parameterizing three-dimensional unstructured data sets over a planar domain. A selection of numerical examples shows the effectiveness of the proposed approach for adaptive spline fitting with (truncated) hierarchical B-spline constructions. In our experiments, improved accuracy is obtained, e.g., from 60% up to 80% for noisy data, while speedups ranging from 4 up to 180 times are observed with respect to classical algorithms. Moreover, our method automatically predicts the local neighborhood graph, leading to much more robust results without the need for delicate free parameter selection. [ABSTRACT FROM AUTHOR]

Assessing changes in protein–protein binding affinity due to mutations helps understanding a wide range of crucial biological processes within cells. Despite significant efforts to create accurate computational models, predicting how mutations affect affinity remains challenging due to the complexity of the biological mechanisms involved. In the present work, a geometric deep learning framework called MuToN is introduced for quantifying protein binding affinity change upon residue mutations. The method, designed with geometric attention networks, is mechanism‐aware. It captures changes in the protein binding interfaces of mutated complexes and assesses the allosteric effects of amino acids. Experimental results highlight MuToN's superiority compared to existing methods. Additionally, MuToN's flexibility and effectiveness are illustrated by its precise predictions of binding affinity changes between SARS‐CoV‐2 variants and the ACE2 complex. [ABSTRACT FROM AUTHOR]

Understanding the interactions between a ligand and its molecular target is crucial in guiding the optimization of molecules for any in silico drug design workflow. Multiple experimental and computational methods have been developed to better understand these intermolecular interactions. With the availability of a large number of structural datasets, there is a need for developing statistical frameworks that improve upon existing physics-based solutions. Here, we report a method based on geometric deep learning that is capable of predicting the binding conformations of ligands to protein targets. A technique to generate graphical representations of proteins was developed to exploit the topological and electrostatic properties of the binding region. The developed framework, based on graph neural networks, learns a statistical potential based on the distance likelihood, which is tailor-made for each ligand–target pair. This potential can be coupled with global optimization algorithms such as differential evolution to reproduce the experimental binding conformations of ligands. We show that the potential based on distance likelihood, described here, performs similarly or better than well-established scoring functions for docking and screening tasks. Overall, this method represents an example of how artificial intelligence can be used to improve structure-based drug design. [ABSTRACT FROM AUTHOR]

The utilization of advanced intraoral scanners to acquire 3D dental models has gained significant popularity in the fields of dentistry and orthodontics. Accurate segmentation and labeling of teeth on digitized 3D dental surface models are crucial for computer-aided treatment planning. At the same time, manual labeling of these models is a time-consuming task. Recent advances in geometric deep learning have demonstrated remarkable efficiency in surface segmentation when applied to raw 3D models. However, segmentation of the dental surface remains challenging due to the atypical and diverse appearance of the patients' teeth. Numerous deep learning methods have been proposed to automate dental surface segmentation. Nevertheless, they still show limitations, particularly in cases where teeth are missing or severely misaligned. To overcome these challenges, we introduce a network operator called dilated edge convolution, which enhances the network's ability to learn additional, more distant features by expanding its receptive field. This leads to improved segmentation results, particularly in complex and challenging cases. To validate the effectiveness of our proposed method, we performed extensive evaluations on the recently published benchmark data set for dental model segmentation Teeth3DS. We compared our approach with several other state-of-the-art methods using a quantitative and qualitative analysis. Through these evaluations, we demonstrate the superiority of our proposed method, showcasing its ability to outperform existing approaches in dental surface segmentation. [ABSTRACT FROM AUTHOR]

Protein–peptide interactions (PPepIs) are vital to understanding cellular functions, which can facilitate the design of novel drugs. As an essential component in forming a PPepI, protein–peptide binding sites are the basis for understanding the mechanisms involved in PPepIs. Therefore, accurately identifying protein–peptide binding sites becomes a critical task. The traditional experimental methods for researching these binding sites are labor-intensive and time-consuming, and some computational tools have been invented to supplement it. However, these computational tools have limitations in generality or accuracy due to the need for ligand information, complex feature construction, or their reliance on modeling based on amino acid residues. To deal with the drawbacks of these computational algorithms, we describe a geometric attention-based network for peptide binding site identification (GAPS) in this work. The proposed model utilizes geometric feature engineering to construct atom representations and incorporates multiple attention mechanisms to update relevant biological features. In addition, the transfer learning strategy is implemented for leveraging the protein–protein binding sites information to enhance the protein–peptide binding sites recognition capability, taking into account the common structure and biological bias between proteins and peptides. Consequently, GAPS demonstrates the state-of-the-art performance and excellent robustness in this task. Moreover, our model exhibits exceptional performance across several extended experiments including predicting the apo protein–peptide, protein–cyclic peptide and the AlphaFold-predicted protein–peptide binding sites. These results confirm that the GAPS model is a powerful, versatile, stable method suitable for diverse binding site predictions. [ABSTRACT FROM AUTHOR]

Urban water management faces new challenges due to the rise of digital solutions and abundant data, leading to the development of data-centric tools for decision-making in global water utilities, with AI technologies poised to become a key trend in the sector. This paper proposes a novel methodology for optimal sensor placement aimed at supporting the creation of a digital twin for water infrastructure. A significant innovation in this study is the creation of a metamodel to estimate pressure at consumption nodes in a water supply system. This metamodel guides the optimal sensor configuration by minimizing the difference between estimated and observed pressures. Our methodology was tested on a synthetic case study, showing accurate results. The estimated pressures at each network node exhibited low error and high accuracy across all sensor configurations tested, highlighting the potential for future development of a digital twin for water distribution systems. [ABSTRACT FROM AUTHOR]

Abstract Assessing changes in protein–protein binding affinity due to mutations helps understanding a wide range of crucial biological processes within cells. Despite significant efforts to create accurate computational models, predicting how mutations affect affinity remains challenging due to the complexity of the biological mechanisms involved. In the present work, a geometric deep learning framework called MuToN is introduced for quantifying protein binding affinity change upon residue mutations. The method, designed with geometric attention networks, is mechanism‐aware. It captures changes in the protein binding interfaces of mutated complexes and assesses the allosteric effects of amino acids. Experimental results highlight MuToN's superiority compared to existing methods. Additionally, MuToN's flexibility and effectiveness are illustrated by its precise predictions of binding affinity changes between SARS‐CoV‐2 variants and the ACE2 complex.

Many complex engineering systems can be represented in a topological form, such as graphs. This paper utilizes a machine learning technique called Geometric Deep Learning (GDL) to aid designers with challenging, graph-centric design problems. The strategy presented here is to take the graph data and apply GDL to seek the best realizable performing solution effectively and efficiently with lower computational costs. This case study used here is the synthesis of analog electrical circuits that attempt to match a specific frequency response within a particular frequency range. Previous studies utilized an enumeration technique to generate 43,249 unique undirected graphs presenting valid potential circuits. Unfortunately, determining the sizing and performance of many circuits can be too expensive. To reduce computational costs with a quantified trade-off in accuracy, the fraction of the circuit graphs and their performance are used as input data to a classification-focused GDL model. Then, the GDL model can be used to predict the remainder cheaply, thus, aiding decision-makers in the search for the best graph solutions. The results discussed in this paper show that additional graph-based features are useful, favorable total set classification accuracy of 80% in using only 10% of the graphs, and iteratively built GDL models can further subdivide the graphs into targeted groups with medians significantly closer to the best and containing 88.2 of the top 100 best-performing graphs on average using 25% of the graphs. [ABSTRACT FROM AUTHOR]

We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure. We classify these automorphic algebras for the case where the homogeneous space has compact stabilisers. This work has applications in the theoretical development of geometric deep learning and also in the theory of automorphic Lie algebras. [ABSTRACT FROM AUTHOR]

Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a universal causal geometric DL framework in which the user specifies a suitable pair of metric spaces X$\mathcal {X}$ and Y$\mathcal {Y}$ and our framework returns a DL model capable of causally approximating any "regular" map sending time series in XZ$\mathcal {X}^{\mathbb {Z}}$ to time series in YZ$\mathcal {Y}^{\mathbb {Z}}$ while respecting their forward flow of information throughout time. Suitable geometries on Y$\mathcal {Y}$ include various (adapted) Wasserstein spaces arising in optimal stopping problems, a variety of statistical manifolds describing the conditional distribution of continuous‐time finite state Markov chains, and all Fréchet spaces admitting a Schauder basis, for example, as in classical finance. Suitable spaces X$\mathcal {X}$ are compact subsets of any Euclidean space. Our results all quantitatively express the number of parameters needed for our DL model to achieve a given approximation error as a function of the target map's regularity and the geometric structure both of X$\mathcal {X}$ and of Y$\mathcal {Y}$. Even when omitting any temporal structure, our universal approximation theorems are the first guarantees that Hölder functions, defined between such X$\mathcal {X}$ and Y$\mathcal {Y}$ can be approximated by DL models. [ABSTRACT FROM AUTHOR]

The use of machine learning in aerodynamic shape optimization problems has significantly increased in recent years. While existing deep learning techniques enable efficient design space exploration on data with an underlying Euclidean or grid-like structure, the direct optimization of non-parametric 3D geometries is still limited. In this article, we propose a geometric deep learning model that generates triangled-based meshed surfaces through the use of a graph variational autoencoder that learns the latent representations of a non-parametric 3D dataset. Once this framework is trained to embed all the input meshes in a properly distributed latent space, its exploration is managed by a genetic algorithm. In this regard, the NSGA-II is the agent in charge of sampling geometries that combine topology and aerodynamic features of the initial ones. Furthermore, in each iteration, it evaluates their aerodynamic performance with CFD in order to guide the optimization process and find the most effective region of the latent space. As a result, those solutions that maximize aerodynamic performance are provided through a Pareto front. The application to a case study and a real-world application is introduced aiming to validate the proposed approach. [ABSTRACT FROM AUTHOR]

Angiogenesis is an essential process in tumorigenesis, tumor invasion, and metastasis, and is an intriguing pathway for drug discovery. Targeting vascular endothelial growth factor receptor 2 (VEGFR2) to inhibit tumor angiogenic pathways has been widely explored and adopted in clinical practice. However, most drugs, such as the Food and Drug Administration –approved drug axitinib (ATC code: L01EK01), have considerable side effects and limited tolerability. Therefore, there is an urgent need for the development of novel VEGFR2 inhibitors. In this study, we propose a novel strategy to design potential candidates targeting VEGFR2 using three-dimensional (3D) deep learning and structural modeling methods. A geometric-enhanced molecular representation learning method (GEM) model employing a graph neural network (GNN) as its underlying predictive algorithm was used to predict the activity of the candidates. In the structural modeling method, flexible docking was performed to screen data with high affinity and explore the mechanism of the inhibitors. Small -molecule compounds with consistently improved properties were identified based on the intersection of the scores obtained from both methods. Candidates identified using the GEM-GNN model were selected for in silico modeling using molecular dynamics simulations to further validate their efficacy. The GEM-GNN model enabled the identification of candidate compounds with potentially more favorable properties than the existing drug, axitinib, while achieving higher efficacy.

Finding new molecules with desirable properties has high computational and overhead costs. Much research has focused on generating candidate molecules in one- and two-dimensional spaces, which has produced some favorable results. However, extending these approaches to molecules in three-dimensional space would be far more useful because the representation of molecules is more realistic, although three-dimensional methods have much higher computational costs. In this work, we developed a geometric deep reinforcement learning agent that generates and optimizes molecules that could interact with a biochemical target. The agent can be used for generating molecules from scratch or for lead optimization when it enhances the properties of a given molecule, whether by enhancing its drug-likeness or increasing its activity toward the target via implicit learning. Thus, the agent works with molecules in three-dimensional space without high computational costs.

Proteins play a central role in biology from immune recognition to brain activity. While major advances in machine learning have improved our ability to predict protein structure from sequence, determining protein function from its sequence or structure remains a major challenge. Here, we introduce holographic convolutional neural network (H-CNN) for proteins, which is a physically motivated machine learning approach to model amino acid preferences in protein structures. H-CNN reflects physical interactions in a protein structure and recapitulates the functional information stored in evolutionary data. H-CNN accurately predicts the impact of mutations on protein stability and binding of protein complexes. Our interpretable computational model for protein structure-function maps could guide design of novel proteins with desired function [ABSTRACT FROM AUTHOR]

G2/M cell cycle checkpoint protein WEE1 kinase is a promising target for inhibiting tumor growth. Although various WEE1 inhibitors have entered clinical investigations, their therapeutic efficacy and safety profile remain unsatisfactory. In this study, we employed a comprehensive virtual screening workflow, which included Schrödinger-Glide molecular docking at different precision levels, as well as the utilization of tools such as MM/GBSA and Deepdock to predict the binding affinity between targets and ligands, in order to identify potential WEE1 inhibitors. Out of ten molecules screened, 50% of these molecules exhibited strong inhibitory activity against WEE1. Among them, compounds 4 and 5 showed excellent inhibitory activity with IC50 values of 1.069 and 3.77 nM respectively, which was comparable to AZD1775. Further investigations revealed that compound 4 displayed significant anti-proliferative effects in A549, PC9, and HuH-7 cells and could also induce apoptosis and G1 phase arrest in PC9 cells. Additionally, molecular dynamics simulations unveiled the binding details of compound 4 with WEE1, notably the crucial hydrogen bond interactions formed with Cys379. In summary, this comprehensive virtual screening workflow, combined with in vitro testing and computational modeling, holds significant importance in the development of promising WEE1 inhibitors. [ABSTRACT FROM AUTHOR]

Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry. [ABSTRACT FROM AUTHOR]

Revealing protein binding sites with other molecules, such as nucleic acids, peptides, or small ligands, sheds light on disease mechanism elucidation and novel drug design. With the explosive growth of proteins in sequence databases, how to accurately and efficiently identify these binding sites from sequences becomes essential. However, current methods mostly rely on expensive multiple sequence alignments or experimental protein structures, limiting their genome-scale applications. Besides, these methods haven’t fully explored the geometry of the protein structures. Here, we propose GPSite, a multi-task network for simultaneously predicting binding residues of DNA, RNA, peptide, protein, ATP, HEM, and metal ions on proteins. GPSite was trained on informative sequence embeddings and predicted structures from protein language models, while comprehensively extracting residual and relational geometric contexts in an end-to-end manner. Experiments demonstrate that GPSite substantially surpasses state-of-the-art sequence-based and structure-based approaches on various benchmark datasets, even when the structures are not well-predicted. The low computational cost of GPSite enables rapid genome-scale binding residue annotations for over 568,000 sequences, providing opportunities to unveil unexplored associations of binding sites with molecular functions, biological processes, and genetic variants. The GPSite webserver and annotation database can be freely accessed at https://bio-web1.nscc-gz.cn/app/GPSite.

Accurate prediction of contacting residue pairs between interacting proteins is very useful for structural characterization of protein–protein interactions. Although significant improvement has been made in inter-protein contact prediction recently, there is still a large room for improving the prediction accuracy. Here we present a new deep learning method referred to as PLMGraph-Inter for inter-protein contact prediction. Specifically, we employ rotationally and translationally invariant geometric graphs obtained from structures of interacting proteins to integrate multiple protein language models, which are successively transformed by graph encoders formed by geometric vector perceptrons and residual networks formed by dimensional hybrid residual blocks to predict inter-protein contacts. Extensive evaluation on multiple test sets illustrates that PLMGraph-Inter outperforms five top inter-protein contact prediction methods, including DeepHomo, GLINTER, CDPred, DeepHomo2, and DRN-1D2D_Inter, by large margins. In addition, we also show that the prediction of PLMGraph-Inter can complement the result of AlphaFold-Multimer. Finally, we show leveraging the contacts predicted by PLMGraph-Inter as constraints for protein–protein docking can dramatically improve its performance for protein complex structure prediction.

Diffusion magnetic resonance imaging is sensitive to the microstructural properties of brain tissue. However, estimating clinically and scientifically relevant microstructural properties from the measured signals remains a highly challenging inverse problem that machine learning may help solve. This study investigated if recently developed rotationally invariant spherical convolutional neural networks can improve microstructural parameter estimation. We trained a spherical convolutional neural network to predict the ground-truth parameter values from efficiently simulated noisy data and applied the trained network to imaging data acquired in a clinical setting to generate microstructural parameter maps. Our network performed better than the spherical mean technique and multi-layer perceptron, achieving higher prediction accuracy than the spherical mean technique with less rotational variance than the multi-layer perceptron. Although we focused on a constrained two-compartment model of neuronal tissue, the network and training pipeline are generalizable and can be used to estimate the parameters of any Gaussian compartment model. To highlight this, we also trained the network to predict the parameters of a three-compartment model that enables the estimation of apparent neural soma density using tensor-valued diffusion encoding.

Neurons can be abstractly represented as skeletons due to the filament nature of neurites. With the rapid development of imaging and image analysis techniques, an increasing amount of neuron skeleton data is being produced. In some scientific studies, it is necessary to dissect the axons and dendrites, which is typically done manually and is both tedious and time-consuming. To automate this process, we have developed a method that relies solely on neuronal skeletons using Geometric Deep Learning (GDL). We demonstrate the effectiveness of this method using pyramidal neurons in mammalian brains, and the results are promising for its application in neuroscience studies. [ABSTRACT FROM AUTHOR]

We present SpecTrHuMS, a Spectral Transformer for 3D triangular Human Mesh Sequence learning which combines known deep learning models with spectral mesh processing to capture characteristics of 3D shapes as well as temporal dependencies between the frames. Unlike previous works in this field, our approach is able to work directly with a compressed representation of the geometry, the spectral coefficients, rather than relying solely on skeleton joints that does not contain surface information. The vertices of each mesh of a sequence are first projected on the eigenvectors of the Graph Laplacian computed from the common triangulation. A convolutional encoder then encodes each frame into lower dimensional latent variables that preserve as much as possible the spectral information. These latent variables are next passed through a transformer architecture so that the model understands the context of the sequence and learns temporal dependencies between the frames. Each frame of the transformer's output is then decoded by a convolutional decoder which aims to reconstruct the input spectral coefficients. Finally, all frames are transformed back into the spatial domain, resulting in a general process able to treat 4D surfaces with a constant connectivity. Our method is evaluated on a prediction task on AMASS, a dataset of human surface sequences, showing the ability of our model to produce realistic movements while preserving the identity of a subject, and showing that this work is a significant step towards efficient and high-quality representation of triangular mesh sequences with constant connectivity. Additional experiments show that our model can be easily extended to other tasks such as long term prediction, completion and that it is generalizable to other datasets with constant connectivity. This work opens up new possibilities for applications in the fields of animation, virtual reality, and computer graphics. Pretrained models, the code to train them and the code to create datasets will be made publicly available. [Display omitted] • This work focuses on generating human triangular mesh sequences. • It consists in the association of a convolutional autoencoder and a transformer. • Spectral coefficients computed from the Graph Laplacian are given as input. • The context of the sequence is understood using a transformer. • The trained model is able to predict movements and preserve the identity of subjects. [ABSTRACT FROM AUTHOR]

Designing realistic digital humans is extremely complex. Most data‐driven generative models used to simplify the creation of their underlying geometric shape do not offer control over the generation of local shape attributes. In this paper, we overcome this limitation by introducing a novel loss function grounded in spectral geometry and applicable to different neural‐network‐based generative models of 3D head and body meshes. Encouraging the latent variables of mesh variational autoencoders (VAEs) or generative adversarial networks (GANs) to follow the local eigenprojections of identity attributes, we improve latent disentanglement and properly decouple the attribute creation. Experimental results show that our local eigenprojection disentangled (LED) models not only offer improved disentanglement with respect to the state‐of‐the‐art, but also maintain good generation capabilities with training times comparable to the vanilla implementations of the models. Our code and pre‐trained models are available at github.com/simofoti/LocalEigenprojDisentangled. [ABSTRACT FROM AUTHOR]