Weitao Du

Weitao Du, Yuanqi Du, Limei Wang et al.

Geometric deep learning enables the encoding of physical symmetries in modeling 3D objects. Despite rapid progress in encoding 3D symmetries into Graph Neural Networks (GNNs), a comprehensive evaluation of the expressiveness of these networks through a local-to-global analysis lacks today. In this paper, we propose a local hierarchy of 3D isomorphism to evaluate the expressive power of equivariant GNNs and investigate the process of representing global geometric information from local patches. Our work leads to two crucial modules for designing expressive and efficient geometric GNNs; namely local substructure encoding (LSE) and frame transition encoding (FTE). To demonstrate the applicability of our theory, we propose LEFTNet which effectively implements these modules and achieves state-of-the-art performance on both scalar-valued and vector-valued molecular property prediction tasks. We further point out the design space for future developments of equivariant graph neural networks. Our codes are available at \url{https://github.com/yuanqidu/LeftNet}.

Arne Schneuing, Charles Harris, Yuanqi Du et al.

Structure-based drug design (SBDD) aims to design small-molecule ligands that bind with high affinity and specificity to pre-determined protein targets. Generative SBDD methods leverage structural data of drugs in complex with their protein targets to propose new drug candidates. These approaches typically place one atom at a time in an autoregressive fashion using the binding pocket as well as previously added ligand atoms as context in each step. Recently a surge of diffusion generative models has entered this domain which hold promise to capture the statistical properties of natural ligands more faithfully. However, most existing methods focus exclusively on bottom-up de novo design of compounds or tackle other drug development challenges with task-specific models. The latter requires curation of suitable datasets, careful engineering of the models and retraining from scratch for each task. Here we show how a single pre-trained diffusion model can be applied to a broader range of problems, such as off-the-shelf property optimization, explicit negative design, and partial molecular design with inpainting. We formulate SBDD as a 3D-conditional generation problem and present DiffSBDD, an SE(3)-equivariant diffusion model that generates novel ligands conditioned on protein pockets. Our in silico experiments demonstrate that DiffSBDD captures the statistics of the ground truth data effectively. Furthermore, we show how additional constraints can be used to improve the generated drug candidates according to a variety of computational metrics. These results support the assumption that diffusion models represent the complex distribution of structural data more accurately than previous methods, and are able to incorporate additional design objectives and constraints changing nothing but the sampling strategy.

Shengchao Liu, Weitao Du, Yanjing Li et al.

Artificial intelligence for scientific discovery has recently generated significant interest within the machine learning and scientific communities, particularly in the domains of chemistry, biology, and material discovery. For these scientific problems, molecules serve as the fundamental building blocks, and machine learning has emerged as a highly effective and powerful tool for modeling their geometric structures. Nevertheless, due to the rapidly evolving process of the field and the knowledge gap between science (e.g., physics, chemistry, & biology) and machine learning communities, a benchmarking study on geometrical representation for such data has not been conducted. To address such an issue, in this paper, we first provide a unified view of the current symmetry-informed geometric methods, classifying them into three main categories: invariance, equivariance with spherical frame basis, and equivariance with vector frame basis. Then we propose a platform, coined Geom3D, which enables benchmarking the effectiveness of geometric strategies. Geom3D contains 16 advanced symmetry-informed geometric representation models and 14 geometric pretraining methods over 46 diverse datasets, including small molecules, proteins, and crystalline materials. We hope that Geom3D can, on the one hand, eliminate barriers for machine learning researchers interested in exploring scientific problems; and, on the other hand, provide valuable guidance for researchers in computational chemistry, structural biology, and materials science, aiding in the informed selection of representation techniques for specific applications.

Shengchao Liu, Divin Yan, Weitao Du et al.

Artificial intelligence models have shown great potential in structure-based drug design, generating ligands with high binding affinities. However, existing models have often overlooked a crucial physical constraint: atoms must maintain a minimum pairwise distance to avoid separation violation, a phenomenon governed by the balance of attractive and repulsive forces. To mitigate such separation violations, we propose NucleusDiff. It models the interactions between atomic nuclei and their surrounding electron clouds by enforcing the distance constraint between the nuclei and manifolds. We quantitatively evaluate NucleusDiff using the CrossDocked2020 dataset and a COVID-19 therapeutic target, demonstrating that NucleusDiff reduces violation rate by up to 100.00% and enhances binding affinity by up to 22.16%, surpassing state-of-the-art models for structure-based drug design. We also provide qualitative analysis through manifold sampling, visually confirming the effectiveness of NucleusDiff in reducing separation violations and improving binding affinities.

Wei Chen, Weitao Du, Zhi-Ming Ma et al.

We study a kind of new SDE that was arisen from the research on optimization in machine learning, we call it power-law dynamic because its stationary distribution cannot have sub-Gaussian tail and obeys power-law. We prove that the power-law dynamic is ergodic with unique stationary distribution, provided the learning rate is small enough. We investigate its first exist time. In particular, we compare the exit times of the (continuous) power-law dynamic and its discretization. The comparison can help guide machine learning algorithm.

Weitao Du, Tao Yang, He Zhang et al.

Diffusion (score-based) generative models have been widely used for modeling various types of complex data, including images, audios, and point clouds. Recently, the deep connection between forward-backward stochastic differential equations (SDEs) and diffusion-based models has been revealed, and several new variants of SDEs are proposed (e.g., sub-VP, critically-damped Langevin) along this line. Despite the empirical success of the hand-crafted fixed forward SDEs, a great quantity of proper forward SDEs remain unexplored. In this work, we propose a general framework for parameterizing the diffusion model, especially the spatial part of the forward SDE. An abstract formalism is introduced with theoretical guarantees, and its connection with previous diffusion models is leveraged. We demonstrate the theoretical advantage of our method from an optimization perspective. Numerical experiments on synthetic datasets, MINIST and CIFAR10 are also presented to validate the effectiveness of our framework.

Tianyu Zhang, Yuxiang Ren, Wenzheng Feng et al.

Unsupervised graph representation learning is a non-trivial topic. The success of contrastive methods in the unsupervised representation learning on structured data inspires similar attempts on the graph. Existing graph contrastive learning (GCL) aims to learn the invariance across multiple augmentation views, which renders it heavily reliant on the handcrafted graph augmentations. However, inappropriate graph data augmentations can potentially jeopardize such invariance. In this paper, we show the potential hazards of inappropriate augmentations and then propose a novel Collaborative Graph Contrastive Learning framework (CGCL). This framework harnesses multiple graph encoders to observe the graph. Features observed from different encoders serve as the contrastive views in contrastive learning, which avoids inducing unstable perturbation and guarantees the invariance. To ensure the collaboration among diverse graph encoders, we propose the concepts of asymmetric architecture and complementary encoders as the design principle. To further prove the rationality, we utilize two quantitative metrics to measure the assembly of CGCL respectively. Extensive experiments demonstrate the advantages of CGCL in unsupervised graph-level representation learning and the potential of collaborative framework. The source code for reproducibility is available at https://github.com/zhangtia16/CGCL

Haorui Li, Weitao Du, Yuqiang Li et al.

Transformer-based autoregressive models have emerged as a unifying paradigm across modalities such as text and images, but their extension to 3D molecule generation remains underexplored. The gap stems from two fundamental challenges: (1) tokenizing molecules into a canonical 1D sequence of tokens that is invariant to both SE(3) transformations and atom index permutations, and (2) designing an architecture capable of modeling hybrid atom-based tokens that couple discrete atom types with continuous 3D coordinates. To address these challenges, we introduce InertialAR. InertialAR devises a canonical tokenization that aligns molecules to their inertial frames and reorders atoms to ensure SE(3) and permutation invariance. Moreover, InertialAR equips the attention mechanism with geometric awareness via geometric rotary positional encoding (GeoRoPE). In addition, it utilizes a hierarchical autoregressive paradigm to predict the next atom-based token, predicting the atom type first and then its 3D coordinates via Diffusion loss. Experimentally, InertialAR achieves state-of-the-art performance on 7 of the 10 evaluation metrics for unconditional molecule generation across QM9, GEOM-Drugs, and B3LYP. Moreover, it significantly outperforms strong baselines in controllable generation for targeted chemical functionality, attaining state-of-the-art results across all 5 metrics.

Weitao Du, Jiujiu Chen, Xuecang Zhang et al.

Recently, artificial intelligence for drug discovery has raised increasing interest in both machine learning and chemistry domains. The fundamental building block for drug discovery is molecule geometry and thus, the molecule's geometrical representation is the main bottleneck to better utilize machine learning techniques for drug discovery. In this work, we propose a pretraining method for molecule joint auto-encoding (MoleculeJAE). MoleculeJAE can learn both the 2D bond (topology) and 3D conformation (geometry) information, and a diffusion process model is applied to mimic the augmented trajectories of such two modalities, based on which, MoleculeJAE will learn the inherent chemical structure in a self-supervised manner. Thus, the pretrained geometrical representation in MoleculeJAE is expected to benefit downstream geometry-related tasks. Empirically, MoleculeJAE proves its effectiveness by reaching state-of-the-art performance on 15 out of 20 tasks by comparing it with 12 competitive baselines.

Shengyin Sun, Wenhao Yu, Yuxiang Ren et al.

Retrosynthesis prediction focuses on identifying reactants capable of synthesizing a target product. Typically, the retrosynthesis prediction involves two phases: Reaction Center Identification and Reactant Generation. However, we argue that most existing methods suffer from two limitations in the two phases: (i) Existing models do not adequately capture the ``face'' information in molecular graphs for the reaction center identification. (ii) Current approaches for the reactant generation predominantly use sequence generation in a 2D space, which lacks versatility in generating reasonable distributions for completed reactive groups and overlooks molecules' inherent 3D properties. To overcome the above limitations, we propose GDiffRetro. For the reaction center identification, GDiffRetro uniquely integrates the original graph with its corresponding dual graph to represent molecular structures, which helps guide the model to focus more on the faces in the graph. For the reactant generation, GDiffRetro employs a conditional diffusion model in 3D to further transform the obtained synthon into a complete reactant. Our experimental findings reveal that GDiffRetro outperforms state-of-the-art semi-template models across various evaluative metrics.

Bangchen Yin, Jiaao Wang, Weitao Du et al.

Molecular dynamics simulations demand an unprecedented combination of accuracy and scalability to tackle grand challenges in catalysis and materials design. To bridge this gap, we present AlphaNet, a local-frame-based equivariant model that simultaneously improves computational efficiency and predictive precision for interatomic interactions. By constructing equivariant local frames with learnable geometric transitions, AlphaNet encodes atomic environments with enhanced representational capacity, achieving state-of-the-art accuracy in energy and force predictions. Extensive benchmarks on large-scale datasets spanning molecular reactions, crystal stability, and surface catalysis (Matbench Discovery and OC2M) demonstrate its superior performance over existing neural network interatomic potentials while ensuring scalability across diverse system sizes with varying types of interatomic interactions. The synergy of accuracy, efficiency, and transferability positions AlphaNet as a transformative tool for modeling multiscale phenomena, decoding dynamics in catalysis and functional interfaces, with direct implications for accelerating the discovery of complex molecular systems and functional materials.

Weitao Du

While standard flow-matching models transport noise to data uniformly, incorporating an explicit generation order - specifically, establishing coarse, low-frequency structure before fine detail - has proven highly effective for synthesizing natural images. Two recent works offer distinct paradigms for this. K-Flow imposes a hard frequency constraint by reinterpreting a frequency scaling variable as flow time, running the trajectory inside a transformed amplitude space. Latent Forcing provides a soft ordering mechanism by coupling the pixel flow with an auxiliary semantic latent flow via asynchronous time schedules, leaving the pixel interpolation path itself untouched. Viewed from the angle of improving pixel generation, we observe that forcing - guiding generation with an earlier-maturing auxiliary stream - offers a highly compatible route to scale-ordered generation without rewriting the core flow coordinate. Building on this, we propose Frequency-Forcing, which realizes K-Flow's frequency ordering through Latent Forcing's soft mechanism: a standard pixel flow is guided by an auxiliary low-frequency stream that matures earlier in time. Unlike Latent Forcing, whose scratchpad relies on a heavy pretrained encoder (e.g., DINO), our frequency scratchpad is derived from the data itself via a lightweight learnable wavelet packet transform. We term this a self-forcing signal, which avoids external dependencies while learning a basis better adapted to data statistics than the fixed bases used in hard frequency flows. On ImageNet-256, Frequency-Forcing consistently improves FID over strong pixel- and latent-space baselines, and naturally composes with a semantic stream to yield further gains. This illustrates that forcing-based scale ordering is a versatile, path-preserving alternative to hard frequency flows.

Kaiwei Zhang, Yange Lin, Guangcheng Wu et al.

The integration of deep learning, particularly AI-Generated Content, with high-quality data derived from ab initio calculations has emerged as a promising avenue for transforming the landscape of scientific research. However, the challenge of designing molecular drugs or materials that incorporate multi-modality prior knowledge remains a critical and complex undertaking. Specifically, achieving a practical molecular design necessitates not only meeting the diversity requirements but also addressing structural and textural constraints with various symmetries outlined by domain experts. In this article, we present an innovative approach to tackle this inverse design problem by formulating it as a multi-modality guidance optimization task. Our proposed solution involves a textural-structure alignment symmetric diffusion framework for the implementation of molecular optimization tasks, namely 3DToMolo. 3DToMolo aims to harmonize diverse modalities including textual description features and graph structural features, aligning them seamlessly to produce molecular structures adhere to specified symmetric structural and textural constraints by experts in the field. Experimental trials across three guidance optimization settings have shown a superior hit optimization performance compared to state-of-the-art methodologies. Moreover, 3DToMolo demonstrates the capability to discover potential novel molecules, incorporating specified target substructures, without the need for prior knowledge. This work not only holds general significance for the advancement of deep learning methodologies but also paves the way for a transformative shift in molecular design strategies. 3DToMolo creates opportunities for a more nuanced and effective exploration of the vast chemical space, opening new frontiers in the development of molecular entities with tailored properties and functionalities.

Weitao Du, Shuning Chang, Jiasheng Tang et al.

In this work, we propose a novel generative learning paradigm, K-Flow, an algorithm that flows along the $K$-amplitude. Here, $k$ is a scaling parameter that organizes frequency bands (or projected coefficients), and amplitude describes the norm of such projected coefficients. By incorporating the $K$-amplitude decomposition, K-Flow enables flow matching across the scaling parameter as time. We discuss three venues and six properties of K-Flow, from theoretical foundations, energy and temporal dynamics, and practical applications, respectively. Specifically, from the practical usage perspective, K-Flow allows steerable generation by controlling the information at different scales. To demonstrate the effectiveness of K-Flow, we conduct experiments on unconditional image generation, class-conditional image generation, and molecule assembly generation. Additionally, we conduct three ablation studies to demonstrate how K-Flow steers scaling parameter to effectively control the resolution of image generation.

Shengchao Liu, Weitao Du, Hannan Xu et al.

In drug discovery, molecular dynamics (MD) simulation for protein-ligand binding provides a powerful tool for predicting binding affinities, estimating transport properties, and exploring pocket sites. There has been a long history of improving the efficiency of MD simulations through better numerical methods and, more recently, by utilizing machine learning (ML) methods. Yet, challenges remain, such as accurate modeling of extended-timescale simulations. To address this issue, we propose NeuralMD, the first ML surrogate that can facilitate numerical MD and provide accurate simulations in protein-ligand binding dynamics. We propose a principled approach that incorporates a novel physics-informed multi-grained group symmetric framework. Specifically, we propose (1) the BindingNet model that satisfies group symmetry using vector frames and captures the multi-level protein-ligand interactions, and (2) an augmented neural differential equation solver that learns the trajectory under Newtonian mechanics. For the experiment, we design ten single-trajectory and three multi-trajectory binding simulation tasks. We demonstrate the efficiency and effectiveness of NeuralMD, achieving over 1K$\times$ speedup compared to standard numerical MD simulations. NeuralMD also outperforms all other ML approaches, achieving up to 15$\times$ reduction in reconstruction error and 70% increase in validity. Additionally, we qualitatively illustrate that the oscillations in the predicted trajectories align more closely with ground-truth dynamics than those of other machine-learning methods. We believe NeuralMD paves the foundation for a new research paradigm in simulating protein-ligand dynamics.

Weitao Du, Shengchao Liu, Xuecang Zhang

By conceiving physical systems as 3D many-body point clouds, geometric graph neural networks (GNNs), such as SE(3)/E(3) equivalent GNNs, have showcased promising performance. In particular, their effective message-passing mechanics make them adept at modeling molecules and crystalline materials. However, current geometric GNNs only offer a mean-field approximation of the many-body system, encapsulated within two-body message passing, thus falling short in capturing intricate relationships within these geometric graphs. To address this limitation, tensor networks, widely employed by computational physics to handle manybody systems using high-order tensors, have been introduced. Nevertheless, integrating these tensorized networks into the message-passing framework of GNNs faces scalability and symmetry conservation (e.g., permutation and rotation) challenges. In response, we introduce an innovative equivariant Matrix Product State (MPS)-based message-passing strategy, through achieving an efficient implementation of the tensor contraction operation. Our method effectively models complex many-body relationships, suppressing mean-field approximations, and captures symmetries within geometric graphs. Importantly, it seamlessly replaces the standard message-passing and layer-aggregation modules intrinsic to geometric GNNs. We empirically validate the superior accuracy of our approach on benchmark tasks, including predicting classical Newton systems and quantum tensor Hamiltonian matrices. To our knowledge, our approach represents the inaugural utilization of parameterized geometric tensor networks.

Shengchao Liu, Weitao Du, Zhiming Ma et al.

Molecule pretraining has quickly become the go-to schema to boost the performance of AI-based drug discovery. Naturally, molecules can be represented as 2D topological graphs or 3D geometric point clouds. Although most existing pertaining methods focus on merely the single modality, recent research has shown that maximizing the mutual information (MI) between such two modalities enhances the molecule representation ability. Meanwhile, existing molecule multi-modal pretraining approaches approximate MI based on the representation space encoded from the topology and geometry, thus resulting in the loss of critical structural information of molecules. To address this issue, we propose MoleculeSDE. MoleculeSDE leverages group symmetric (e.g., SE(3)-equivariant and reflection-antisymmetric) stochastic differential equation models to generate the 3D geometries from 2D topologies, and vice versa, directly in the input space. It not only obtains tighter MI bound but also enables prosperous downstream tasks than the previous work. By comparing with 17 pretraining baselines, we empirically verify that MoleculeSDE can learn an expressive representation with state-of-the-art performance on 26 out of 32 downstream tasks.

Weitao Du, He Zhang, Yuanqi Du et al.

Group equivariance (e.g. SE(3) equivariance) is a critical physical symmetry in science, from classical and quantum physics to computational biology. It enables robust and accurate prediction under arbitrary reference transformations. In light of this, great efforts have been put on encoding this symmetry into deep neural networks, which has been shown to improve the generalization performance and data efficiency for downstream tasks. Constructing an equivariant neural network generally brings high computational costs to ensure expressiveness. Therefore, how to better trade-off the expressiveness and computational efficiency plays a core role in the design of the equivariant deep learning models. In this paper, we propose a framework to construct SE(3) equivariant graph neural networks that can approximate the geometric quantities efficiently. Inspired by differential geometry and physics, we introduce equivariant local complete frames to graph neural networks, such that tensor information at given orders can be projected onto the frames. The local frame is constructed to form an orthonormal basis that avoids direction degeneration and ensure completeness. Since the frames are built only by cross product operations, our method is computationally efficient. We evaluate our method on two tasks: Newton mechanics modeling and equilibrium molecule conformation generation. Extensive experimental results demonstrate that our model achieves the best or competitive performance in two types of datasets.

Wei Huang, Yayong Li, Weitao Du et al.

Graph convolutional networks (GCNs) and their variants have achieved great success in dealing with graph-structured data. Nevertheless, it is well known that deep GCNs suffer from the over-smoothing problem, where node representations tend to be indistinguishable as more layers are stacked up. The theoretical research to date on deep GCNs has focused primarily on expressive power rather than trainability, an optimization perspective. Compared to expressivity, trainability attempts to address a more fundamental question: Given a sufficiently expressive space of models, can we successfully find a good solution via gradient descent-based optimizers? This work fills this gap by exploiting the Graph Neural Tangent Kernel (GNTK), which governs the optimization trajectory under gradient descent for wide GCNs. We formulate the asymptotic behaviors of GNTK in the large depth, which enables us to reveal the dropping trainability of wide and deep GCNs at an exponential rate in the optimization process. Additionally, we extend our theoretical framework to analyze residual connection-based techniques, which are found to be merely able to mitigate the exponential decay of trainability mildly. Inspired by our theoretical insights on trainability, we propose Critical DropEdge, a connectivity-aware and graph-adaptive sampling method, to alleviate the exponential decay problem more fundamentally. Experimental evaluation consistently confirms using our proposed method can achieve better results compared to relevant counterparts with both infinite-width and finite-width.

Wei Huang, Weitao Du, Richard Yi Da Xu et al.

Most theoretical studies explaining the regularization effect in deep learning have only focused on gradient descent with a sufficient small learning rate or even gradient flow (infinitesimal learning rate). Such researches, however, have neglected a reasonably large learning rate applied in most practical applications. In this work, we characterize the implicit bias effect of deep linear networks for binary classification using the logistic loss in the large learning rate regime, inspired by the seminal work by Lewkowycz et al. [26] in a regression setting with squared loss. They found a learning rate regime with a large stepsize named the catapult phase, where the loss grows at the early stage of training and eventually converges to a minimum that is flatter than those found in the small learning rate regime. We claim that depending on the separation conditions of data, the gradient descent iterates will converge to a flatter minimum in the catapult phase. We rigorously prove this claim under the assumption of degenerate data by overcoming the difficulty of the non-constant Hessian of logistic loss and further characterize the behavior of loss and Hessian for non-separable data. Finally, we demonstrate that flatter minima in the space spanned by non-separable data along with the learning rate in the catapult phase can lead to better generalization empirically.

Wei Huang, Weitao Du, Richard Yi Da Xu

The prevailing thinking is that orthogonal weights are crucial to enforcing dynamical isometry and speeding up training. The increase in learning speed that results from orthogonal initialization in linear networks has been well-proven. However, while the same is believed to also hold for nonlinear networks when the dynamical isometry condition is satisfied, the training dynamics behind this contention have not been thoroughly explored. In this work, we study the dynamics of ultra-wide networks across a range of architectures, including Fully Connected Networks (FCNs) and Convolutional Neural Networks (CNNs) with orthogonal initialization via neural tangent kernel (NTK). Through a series of propositions and lemmas, we prove that two NTKs, one corresponding to Gaussian weights and one to orthogonal weights, are equal when the network width is infinite. Further, during training, the NTK of an orthogonally-initialized infinite-width network should theoretically remain constant. This suggests that the orthogonal initialization cannot speed up training in the NTK (lazy training) regime, contrary to the prevailing thoughts. In order to explore under what circumstances can orthogonality accelerate training, we conduct a thorough empirical investigation outside the NTK regime. We find that when the hyper-parameters are set to achieve a linear regime in nonlinear activation, orthogonal initialization can improve the learning speed with a large learning rate or large depth.

Wei Huang, Richard Yi Da Xu, Weitao Du et al.

In recent years, the mean field theory has been applied to the study of neural networks and has achieved a great deal of success. The theory has been applied to various neural network structures, including CNNs, RNNs, Residual networks, and Batch normalization. Inevitably, recent work has also covered the use of dropout. The mean field theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. However, the gradient backpropagation is derived under the gradient independence assumption that weights used during feed forward are drawn independently from the ones used in backpropagation. This is not how neural networks are trained in a real setting. Instead, the same weights used in a feed-forward step needs to be carried over to its corresponding backpropagation. Using this realistic condition, we perform theoretical computation on linear dropout networks and a series of experiments on dropout networks. Our empirical results show an interesting phenomenon that the length gradients can backpropagate for a single input and a pair of inputs are governed by the same depth scale. Besides, we study the relationship between variance and mean of statistical metrics of the gradient and shown an emergence of universality. Finally, we investigate the maximum trainable length for deep dropout networks through a series of experiments using MNIST and CIFAR10 and provide a more precise empirical formula that describes the trainable length than original work.