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Minkai Xu

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NeurIPS Conference 2025 Conference Paper

3D Interaction Geometric Pre-training for Molecular Relational Learning

  • Namkyeong Lee
  • Yunhak Oh
  • Heewoong Noh
  • Gyoung S. Na
  • Minkai Xu
  • Hanchen Wang
  • Tianfan Fu
  • Chanyoung Park

Molecular Relational Learning (MRL) is a rapidly growing field that focuses on understanding the interaction dynamics between molecules, which is crucial for applications ranging from catalyst engineering to drug discovery. Despite recent progress, earlier MRL approaches are limited to using only the 2D topological structure of molecules, as obtaining the 3D interaction geometry remains prohibitively expensive. This paper introduces a novel 3D geometric pre-training strategy for MRL (3DMRL) that incorporates a 3D virtual interaction environment, overcoming the limitations of costly traditional quantum mechanical calculation methods. With the constructed 3D virtual interaction environment, 3DMRL trains 2D MRL model to learn the global and local 3D geometric information of molecular interaction. Extensive experiments on various tasks using real-world datasets, including out-of-distribution and extrapolation scenarios, demonstrate the effectiveness of 3DMRL, showing up to a 24. 93% improvement in performance across 40 tasks. Our code is publicly available at https: //github. com/Namkyeong/3DMRL.

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ICLR Conference 2025 Conference Paper

Energy-Based Diffusion Language Models for Text Generation

  • Minkai Xu
  • Tomas Geffner
  • Karsten Kreis
  • Weili Nie
  • Yilun Xu
  • Jure Leskovec
  • Stefano Ermon
  • Arash Vahdat

Despite remarkable progress in autoregressive language models, alternative generative paradigms beyond left-to-right generation are still being actively explored. Discrete diffusion models, with the capacity for parallel generation, have recently emerged as a promising alternative. Unfortunately, these models still underperform the autoregressive counterparts, with the performance gap increasing when reducing the number of sampling steps. Our analysis reveals that this degradation is a consequence of an imperfect approximation used by diffusion models. In this work, we propose Energy-based Diffusion Language Model (EDLM), an energy-based model operating at the full sequence level for each diffusion step, introduced to improve the underlying approximation used by diffusion models. More specifically, we introduce an EBM in a residual form, and show that its parameters can be obtained by leveraging a pretrained autoregressive model or by finetuning a bidirectional transformer via noise contrastive estimation. We also propose an efficient generation algorithm via parallel important sampling. Comprehensive experiments on language modeling benchmarks show that our model can consistently outperform state-of-the-art diffusion models by a significant margin, and approaches autoregressive models' perplexity. We further show that, without any generation performance drop, our framework offers a 1.3x sampling speedup over existing diffusion models. Reproduced code is available at https://github.com/MinkaiXu/Energy-Diffusion-LLM.

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ICML Conference 2025 Conference Paper

Smooth Interpolation for Improved Discrete Graph Generative Models

  • Yuxuan Song
  • Juntong Shi
  • Jingjing Gong
  • Minkai Xu
  • Stefano Ermon
  • Hao Zhou 0012
  • Wei-Ying Ma

Though typically represented by the discrete node and edge attributes, the graph topological information can be sufficiently captured by the graph spectrum in a continuous space. It is believed that incorporating the continuity of graph topological information into the generative process design could establish a superior paradigm for graph generative modeling. Motivated by such prior and recent advancements in the generative paradigm, we propose Graph Bayesian Flow Networks (GraphBFN) in this paper, a principled generative framework that designs an alternative generative process emphasizing the dynamics of topological information. Unlike recent discrete-diffusion-based methods, GraphBFNemploys the continuous counts derived from sampling infinite times from a categorical distribution as latent to facilitate a smooth decomposition of topological information, demonstrating enhanced effectiveness. To effectively realize the concept, we further develop an advanced sampling strategy and new time-scheduling techniques to overcome practical barriers and boost performance. Through extensive experimental validation on both generic graph and molecular graph generation tasks, GraphBFN could consistently achieve superior or competitive performance with significantly higher training and sampling efficiency.

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ICLR Conference 2025 Conference Paper

SuperCorrect: Advancing Small LLM Reasoning with Thought Template Distillation and Self-Correction

  • Ling Yang 0006
  • Zhaochen Yu
  • Tianjun Zhang
  • Minkai Xu
  • Joseph E. Gonzalez
  • Bin Cui 0001
  • Shuicheng Yan

Large language models (LLMs) like GPT-4, DeepSeek-R1, and ReasonFlux have shown significant improvements in various reasoning tasks. However, smaller LLMs still struggle with complex mathematical reasoning because they fail to effectively identify and correct reasoning errors. Recent reflection-based methods aim to address these issues by enabling self-reflection and self-correction, but they still face challenges in independently detecting errors in their reasoning steps. To overcome these limitations, we propose SuperCorrect, a novel two-stage framework that uses a large teacher model to supervise and correct both the reasoning and reflection processes of a smaller student model. In the first stage, we extract hierarchical high-level and detailed thought templates from the teacher model to guide the student model in eliciting more fine-grained reasoning thoughts. In the second stage, we introduce cross-model collaborative direct preference optimization (DPO) to enhance the self-correction abilities of the student model by following the teacher's correction traces during training. This cross-model DPO approach teaches the student model to effectively locate and resolve erroneous thoughts with error-driven insights from the teacher model, breaking the bottleneck of its thoughts and acquiring new skills and knowledge to tackle challenging problems. Extensive experiments consistently demonstrate our superiority over previous methods. Notably, our SuperCorrect-7B model significantly surpasses powerful DeepSeekMath-7B by 7.8\%/5.3\% and Qwen2.5-Math-7B by 15.1\%/6.3\% on MATH/GSM8K benchmarks, achieving new SOTA performance among all 7B models. Code is available at: https://github.com/YangLing0818/SuperCorrect-llm

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ICLR Conference 2025 Conference Paper

TabDiff: a Mixed-type Diffusion Model for Tabular Data Generation

  • Juntong Shi
  • Minkai Xu
  • Harper Hua
  • Hengrui Zhang
  • Stefano Ermon
  • Jure Leskovec

Synthesizing high-quality tabular data is an important topic in many data science tasks, ranging from dataset augmentation to privacy protection. However, developing expressive generative models for tabular data is challenging due to its inherent heterogeneous data types, complex inter-correlations, and intricate column-wise distributions. In this paper, we introduce TabDiff, a joint diffusion framework that models all mixed-type distributions of tabular data in one model. Our key innovation is the development of a joint continuous-time diffusion process for numerical and categorical data, where we propose feature-wise learnable diffusion processes to counter the high disparity of different feature distributions. TabDiff is parameterized by a transformer handling different input types, and the entire framework can be efficiently optimized in an end-to-end fashion. We further introduce a mixed-type stochastic sampler to automatically correct the accumulated decoding error during sampling, and propose classifier-free guidance for conditional missing column value imputation. Comprehensive experiments on seven datasets demonstrate that TabDiff achieves superior average performance over existing competitive baselines across all eight metrics, with up to $22.5\%$ improvement over the state-of-the-art model on pair-wise column correlation estimations. Code is available at https://github.com/MinkaiXu/TabDiff.

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NeurIPS Conference 2024 Conference Paper

Aligning Target-Aware Molecule Diffusion Models with Exact Energy Optimization

  • Siyi Gu
  • Minkai Xu
  • Alexander Powers
  • Weili Nie
  • Tomas Geffner
  • Karsten Kreis
  • Jure Leskovec
  • Arash Vahdat

Generating ligand molecules for specific protein targets, known as structure-based drug design, is a fundamental problem in therapeutics development and biological discovery. Recently, target-aware generative models, especially diffusion models, have shown great promise in modeling protein-ligand interactions and generating candidate drugs. However, existing models primarily focus on learning the chemical distribution of all drug candidates, which lacks effective steerability on the chemical quality of model generations. In this paper, we propose a novel and general alignment framework to align pretrained target diffusion models with preferred functional properties, named AliDiff. AliDiff shifts the target-conditioned chemical distribution towards regions with higher binding affinity and structural rationality, specified by user-defined reward functions, via the preference optimization approach. To avoid the overfitting problem in common preference optimization objectives, we further develop an improved Exact Energy Preference Optimization method to yield an exact and efficient alignment of the diffusion models, and provide the closed-form expression for the converged distribution. Empirical studies on the CrossDocked2020 benchmark show that AliDiff can generate molecules with state-of-the-art binding energies with up to -7. 07 Avg. Vina Score, while maintaining strong molecular properties. Code is available at https: //github. com/MinkaiXu/AliDiff.

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NeurIPS Conference 2024 Conference Paper

Buffer of Thoughts: Thought-Augmented Reasoning with Large Language Models

  • Ling Yang
  • Zhaochen Yu
  • Tianjun Zhang
  • Shiyi Cao
  • Minkai Xu
  • Wentao Zhang
  • Joseph E. Gonzalez
  • Bin Cui

We introduce Buffer of Thoughts (BoT), a novel and versatile thought-augmented reasoning approach for enhancing accuracy, efficiency and robustness of large language models (LLMs). Specifically, we propose meta-buffer to store a series of informative high-level thoughts, namely thought-template, distilled from the problem-solving processes across various tasks. Then for each problem, we retrieve a relevant thought-template and adaptively instantiate it with specific reasoning structures to conduct efficient reasoning. To guarantee the scalability and stability, we further propose buffer-manager to dynamically update the meta-buffer, thus enhancing the capacity of meta-buffer as more tasks are solved. We conduct extensive experiments on 10 challenging reasoning-intensive tasks, and achieve significant performance improvements over previous SOTA methods: 11\% on Game of 24, 20\% on Geometric Shapes and 51\% on Checkmate-in-One. Further analysis demonstrate the superior generalization ability and model robustness of our BoT, while requiring only 12\% of the cost of multi-query prompting methods (e. g. , tree/graph of thoughts) on average. Code is available at: https: //github. com/YangLing0818/buffer-of-thought-llm

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ICLR Conference 2024 Conference Paper

Cross-Modal Contextualized Diffusion Models for Text-Guided Visual Generation and Editing

  • Ling Yang 0006
  • Zhilong Zhang
  • Zhaochen Yu
  • Jingwei Liu
  • Minkai Xu
  • Stefano Ermon
  • Bin Cui 0001

Conditional diffusion models have exhibited superior performance in high-fidelity text-guided visual generation and editing. Nevertheless, prevailing text-guided visual diffusion models primarily focus on incorporating text-visual relationships exclusively into the reverse process, often disregarding their relevance in the forward process. This inconsistency between forward and reverse processes may limit the precise conveyance of textual semantics in visual synthesis results. To address this issue, we propose a novel and general contextualized diffusion model (ContextDiff) by incorporating the cross-modal context encompassing interactions and alignments between text condition and visual sample into forward and reverse processes. We propagate this context to all timesteps in the two processes to adapt their trajectories, thereby facilitating cross-modal conditional modeling. We generalize our contextualized diffusion to both DDPMs and DDIMs with theoretical derivations, and demonstrate the effectiveness of our model in evaluations with two challenging tasks: text-to-image generation, and text-to-video editing. In each task, our ContextDiff achieves new state-of-the-art performance, significantly enhancing the semantic alignment between text condition and generated samples, as evidenced by quantitative and qualitative evaluations. Our code is available at https://github.com/YangLing0818/ContextDiff

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ICML Conference 2024 Conference Paper

Equivariant Graph Neural Operator for Modeling 3D Dynamics

  • Minkai Xu
  • Jiaqi Han
  • Aaron Lou
  • Jean Kossaifi
  • Arvind Ramanathan
  • Kamyar Azizzadenesheli
  • Jure Leskovec
  • Stefano Ermon

Modeling the complex three-dimensional (3D) dynamics of relational systems is an important problem in the natural sciences, with applications ranging from molecular simulations to particle mechanics. Machine learning methods have achieved good success by learning graph neural networks to model spatial interactions. However, these approaches do not faithfully capture temporal correlations since they only model next-step predictions. In this work, we propose Equivariant Graph Neural Operator (EGNO), a novel and principled method that directly models dynamics as trajectories instead of just next-step prediction. Different from existing methods, EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. To capture the temporal correlations while keeping the intrinsic SE(3)-equivariance, we develop equivariant temporal convolutions parameterized in the Fourier space and build EGNO by stacking the Fourier layers over equivariant networks. EGNO is the first operator learning framework that is capable of modeling solution dynamics functions over time while retaining 3D equivariance. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods, thanks to the equivariant temporal modeling. Our code is available at https: //github. com/MinkaiXu/egno.

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NeurIPS Conference 2024 Conference Paper

Geometric Trajectory Diffusion Models

  • Jiaqi Han
  • Minkai Xu
  • Aaron Lou
  • Haotian Ye
  • Stefano Ermon

Generative models have shown great promise in generating 3D geometric systems, which is a fundamental problem in many natural science domains such as molecule and protein design. However, existing approaches only operate on static structures, neglecting the fact that physical systems are always dynamic in nature. In this work, we propose geometric trajectory diffusion models (GeoTDM), the first diffusion model for modeling the temporal distribution of 3D geometric trajectories. Modeling such distribution is challenging as it requires capturing both the complex spatial interactions with physical symmetries and temporal correspondence encapsulated in the dynamics. We theoretically justify that diffusion models with equivariant temporal kernels can lead to density with desired symmetry, and develop a novel transition kernel leveraging SE(3)-equivariant spatial convolution and temporal attention. Furthermore, to induce an expressive trajectory distribution for conditional generation, we introduce a generalized learnable geometric prior into the forward diffusion process to enhance temporal conditioning. We conduct extensive experiments on both unconditional and conditional generation in various scenarios, including physical simulation, molecular dynamics, and pedestrian motion. Empirical results on a wide suite of metrics demonstrate that GeoTDM can generate realistic geometric trajectories with significantly higher quality.

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NeurIPS Conference 2024 Conference Paper

MADiff: Offline Multi-agent Learning with Diffusion Models

  • Zhengbang Zhu
  • Minghuan Liu
  • Liyuan Mao
  • Bingyi Kang
  • Minkai Xu
  • Yong Yu
  • Stefano Ermon
  • Weinan Zhang

Offline reinforcement learning (RL) aims to learn policies from pre-existing datasets without further interactions, making it a challenging task. Q-learning algorithms struggle with extrapolation errors in offline settings, while supervised learning methods are constrained by model expressiveness. Recently, diffusion models (DMs) have shown promise in overcoming these limitations in single-agent learning, but their application in multi-agent scenarios remains unclear. Generating trajectories for each agent with independent DMs may impede coordination, while concatenating all agents’ information can lead to low sample efficiency. Accordingly, we propose MADiff, which is realized with an attention-based diffusion model to model the complex coordination among behaviors of multiple agents. To our knowledge, MADiff is the first diffusion-based multi-agent learning framework, functioning as both a decentralized policy and a centralized controller. During decentralized executions, MADiff simultaneously performs teammate modeling, and the centralized controller can also be applied in multi-agent trajectory predictions. Our experiments demonstrate that MADiff outperforms baseline algorithms across various multi-agent learning tasks, highlighting its effectiveness in modeling complex multi-agent interactions.

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ICML Conference 2024 Conference Paper

Mastering Text-to-Image Diffusion: Recaptioning, Planning, and Generating with Multimodal LLMs

  • Ling Yang 0006
  • Zhaochen Yu
  • Chenlin Meng
  • Minkai Xu
  • Stefano Ermon
  • Bin Cui 0001

Diffusion models have exhibit exceptional performance in text-to-image generation and editing. However, existing methods often face challenges when handling complex text prompts that involve multiple objects with multiple attributes and relationships. In this paper, we propose a brand new training-free text-to-image generation/editing framework, namely Recaption, Plan and Generate (RPG), harnessing the powerful chain-of-thought reasoning ability of multimodal LLMs to enhance the compositionality of text-to-image diffusion models. Our approach employs the MLLM as a global planner to decompose the process of generating complex images into multiple simpler generation tasks within subregions. We propose complementary regional diffusion to enable region-wise compositional generation. Furthermore, we integrate text-guided image generation and editing within the proposed RPG in a closed-loop fashion, thereby enhancing generalization ability. Extensive experiments demonstrate our RPG outperforms state-of-the-art text-to-image models, including DALL-E 3 and SDXL, particularly in multi-category object composition and text-image semantic alignment. Notably, our RPG framework exhibits wide compatibility with various MLLM architectures and diffusion backbones. Our code is available at https: //github. com/YangLing0818/RPG-DiffusionMaster

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NeurIPS Conference 2024 Conference Paper

RealCompo: Balancing Realism and Compositionality Improves Text-to-Image Diffusion Models

  • Xinchen Zhang
  • Ling Yang
  • Yaqi Cai
  • Zhaochen Yu
  • Kai-Ni Wang
  • Jiake Xie
  • Ye Tian
  • Minkai Xu

Diffusion models have achieved remarkable advancements in text-to-image generation. However, existing models still have many difficulties when faced with multiple-object compositional generation. In this paper, we propose RealCompo, a new training-free and transferred-friendly text-to-image generation framework, which aims to leverage the respective advantages of text-to-image models and spatial-aware image diffusion models (e. g. , layout, keypoints and segmentation maps) to enhance both realism and compositionality of the generated images. An intuitive and novel balancer is proposed to dynamically balance the strengths of the two models in denoising process, allowing plug-and-play use of any model without extra training. Extensive experiments show that our RealCompo consistently outperforms state-of-the-art text-to-image models and spatial-aware image diffusion models in multiple-object compositional generation while keeping satisfactory realism and compositionality of the generated images. Notably, our RealCompo can be seamlessly extended with a wide range of spatial-aware image diffusion models and stylized diffusion models. Code is available at: https: //github. com/YangLing0818/RealCompo

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NeurIPS Conference 2024 Conference Paper

TFG: Unified Training-Free Guidance for Diffusion Models

  • Haotian Ye
  • Haowei Lin
  • Jiaqi Han
  • Minkai Xu
  • Sheng Liu
  • Yitao Liang
  • Jianzhu Ma
  • James Zou

Given an unconditional diffusion model and a predictor for a target property of interest (e. g. , a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. Existing methods, though effective in various individual applications, often lack theoretical grounding and rigorous testing on extensive benchmarks. As a result, they could even fail on simple tasks, and applying them to a new problem becomes unavoidably difficult. This paper introduces a novel algorithmic framework encompassing existing methods as special cases, unifying the study of training-free guidance into the analysis of an algorithm-agnostic design space. Via theoretical and empirical investigation, we propose an efficient and effective hyper-parameter searching strategy that can be readily applied to any downstream task. We systematically benchmark across 7 diffusion models on 16 tasks with 40 targets, and improve performance by 8. 5% on average. Our framework and benchmark offer a solid foundation for conditional generation in a training-free manner.

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ICLR Conference 2024 Conference Paper

VQGraph: Rethinking Graph Representation Space for Bridging GNNs and MLPs

  • Ling Yang 0006
  • Ye Tian
  • Minkai Xu
  • Zhongyi Liu 0001
  • Shenda Hong
  • Wei Qu
  • Wentao Zhang 0001
  • Bin Cui 0001

GNN-to-MLP distillation aims to utilize knowledge distillation (KD) to learn computationally-efficient multi-layer perceptron (student MLP) on graph data by mimicking the output representations of teacher GNN. Existing methods mainly make the MLP to mimic the GNN predictions over a few class labels. However, the class space may not be expressive enough for covering numerous diverse local graph structures, thus limiting the performance of knowledge transfer from GNN to MLP. To address this issue, we propose to learn a new powerful graph representation space by directly labeling nodes' diverse local structures for GNN-to-MLP distillation. Specifically, we propose a variant of VQ-VAE to learn a structure-aware tokenizer on graph data that can encode each node's local substructure as a discrete code. The discrete codes constitute a codebook as a new graph representation space that is able to identify different local graph structures of nodes with the corresponding code indices. Then, based on the learned codebook, we propose a new distillation target, namely soft code assignments, to directly transfer the structural knowledge of each node from GNN to MLP. The resulting framework VQGraph achieves new state-of-the-art performance on GNN-to-MLP distillation in both transductive and inductive settings across seven graph datasets. We show that VQGraph with better performance infers faster than GNNs by 828×, and also achieves accuracy improvement over GNNs and stand-alone MLPs by 3.90% and 28.05% on average, respectively. Our code is available at https://github.com/YangLing0818/VQGraph

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ICML Conference 2023 Conference Paper

Coarse-to-Fine: a Hierarchical Diffusion Model for Molecule Generation in 3D

  • Bo Qiang
  • Yuxuan Song
  • Minkai Xu
  • Jingjing Gong
  • Bowen Gao
  • Hao Zhou 0012
  • Wei-Ying Ma
  • Yanyan Lan

Generating desirable molecular structures in 3D is a fundamental problem for drug discovery. Despite the considerable progress we have achieved, existing methods usually generate molecules in atom resolution and ignore intrinsic local structures such as rings, which leads to poor quality in generated structures, especially when generating large molecules. Fragment-based molecule generation is a promising strategy, however, it is nontrivial to be adapted for 3D non-autoregressive generations because of the combinational optimization problems. In this paper, we utilize a coarse-to-fine strategy to tackle this problem, in which a Hierarchical Diffusion-based model (i. e. HierDiff) is proposed to preserve the validity of local segments without relying on autoregressive modeling. Specifically, HierDiff first generates coarse-grained molecule geometries via an equivariant diffusion process, where each coarse-grained node reflects a fragment in a molecule. Then the coarse-grained nodes are decoded into fine-grained fragments by a message-passing process and a newly designed iterative refined sampling module. Lastly, the fine-grained fragments are then assembled to derive a complete atomic molecular structure. Extensive experiments demonstrate that HierDiff consistently improves the quality of molecule generation over existing methods.

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NeurIPS Conference 2023 Conference Paper

Equivariant Flow Matching with Hybrid Probability Transport for 3D Molecule Generation

  • Yuxuan Song
  • Jingjing Gong
  • Minkai Xu
  • Ziyao Cao
  • Yanyan Lan
  • Stefano Ermon
  • Hao Zhou
  • Wei-Ying Ma

The generation of 3D molecules requires simultaneously deciding the categorical features (atom types) and continuous features (atom coordinates). Deep generative models, especially Diffusion Models (DMs), have demonstrated effectiveness in generating feature-rich geometries. However, existing DMs typically suffer from unstable probability dynamics with inefficient sampling speed. In this paper, we introduce geometric flow matching, which enjoys the advantages of both equivariant modeling and stabilized probability dynamics. More specifically, we propose a hybrid probability path where the coordinates probability path is regularized by an equivariant optimal transport, and the information between different modalities is aligned. Experimentally, the proposed method could consistently achieve better performance on multiple molecule generation benchmarks with 4. 75$\times$ speed up of sampling on average.

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ICML Conference 2023 Conference Paper

FusionRetro: Molecule Representation Fusion via In-Context Learning for Retrosynthetic Planning

  • Songtao Liu
  • Zhengkai Tu
  • Minkai Xu
  • Zuobai Zhang
  • Lu Lin 0001
  • Rex Ying
  • Jian Tang 0005
  • Peilin Zhao

Retrosynthetic planning aims to devise a complete multi-step synthetic route from starting materials to a target molecule. Current strategies use a decoupled approach of single-step retrosynthesis models and search algorithms, taking only the product as the input to predict the reactants for each planning step and ignoring valuable context information along the synthetic route. In this work, we propose a novel framework that utilizes context information for improved retrosynthetic planning. We view synthetic routes as reaction graphs and propose to incorporate context through three principled steps: encode molecules into embeddings, aggregate information over routes, and readout to predict reactants. Our approach is the first attempt to utilize in-context learning for retrosynthesis prediction in retrosynthetic planning. The entire framework can be efficiently optimized in an end-to-end fashion and produce more practical and accurate predictions. Comprehensive experiments demonstrate that by fusing in the context information over routes, our model significantly improves the performance of retrosynthetic planning over baselines that are not context-aware, especially for long synthetic routes. Code is available at https: //github. com/SongtaoLiu0823/FusionRetro.

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ICML Conference 2023 Conference Paper

Geometric Latent Diffusion Models for 3D Molecule Generation

  • Minkai Xu
  • Alexander S. Powers
  • Ron O. Dror
  • Stefano Ermon
  • Jure Leskovec

Generative models, especially diffusion models (DMs), have achieved promising results for generating feature-rich geometries and advancing foundational science problems such as molecule design. Inspired by the recent huge success of Stable (latent) Diffusion models, we propose a novel and principled method for 3D molecule generation named Geometric Latent Diffusion Models (GeoLDM). GeoLDM is the first latent DM model for the molecular geometry domain, composed of autoencoders encoding structures into continuous latent codes and DMs operating in the latent space. Our key innovation is that for modeling the 3D molecular geometries, we capture its critical roto-translational equivariance constraints by building a point-structured latent space with both invariant scalars and equivariant tensors. Extensive experiments demonstrate that GeoLDM can consistently achieve better performance on multiple molecule generation benchmarks, with up to 7% improvement for the valid percentage of large biomolecules. Results also demonstrate GeoLDM’s higher capacity for controllable generation thanks to the latent modeling. Code is provided at https: //github. com/MinkaiXu/GeoLDM.

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NeurIPS Conference 2023 Conference Paper

Scaling Riemannian Diffusion Models

  • Aaron Lou
  • Minkai Xu
  • Adam Farris
  • Stefano Ermon

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement and is competitive with other techniques. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds, including $SU(n)$ lattices in the context of lattice quantum chromodynamics (QCD). Finally, we apply our models to contrastively learned hyperspherical embeddings, curbing the representation collapse problem in the projection head and closing the gap between theory and practice.

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NeurIPS Conference 2023 Conference Paper

When Do Graph Neural Networks Help with Node Classification? Investigating the Homophily Principle on Node Distinguishability

  • Sitao Luan
  • Chenqing Hua
  • Minkai Xu
  • Qincheng Lu
  • Jiaqi Zhu
  • Xiao-Wen Chang
  • Jie Fu
  • Jure Leskovec

Homophily principle, i. e. , nodes with the same labels are more likely to be connected, has been believed to be the main reason for the performance superiority of Graph Neural Networks (GNNs) over Neural Networks on node classification tasks. Recent research suggests that, even in the absence of homophily, the advantage of GNNs still exists as long as nodes from the same class share similar neighborhood patterns. However, this argument only considers intra-class Node Distinguishability (ND) but neglects inter-class ND, which provides incomplete understanding of homophily on GNNs. In this paper, we first demonstrate such deficiency with examples and argue that an ideal situation for ND is to have smaller intra-class ND than inter-class ND. To formulate this idea and study ND deeply, we propose Contextual Stochastic Block Model for Homophily (CSBM-H) and define two metrics, Probabilistic Bayes Error (PBE) and negative generalized Jeffreys divergence, to quantify ND. With the metrics, we visualize and analyze how graph filters, node degree distributions and class variances influence ND, and investigate the combined effect of intra- and inter-class ND. Besides, we discovered the mid-homophily pitfall, which occurs widely in graph datasets. Furthermore, we verified that, in real-work tasks, the superiority of GNNs is indeed closely related to both intra- and inter-class ND regardless of homophily levels. Grounded in this observation, we propose a new hypothesis-testing based performance metric beyond homophily, which is non-linear, feature-based and can provide statistical threshold value for GNNs' the superiority. Experiments indicate that it is significantly more effective than the existing homophily metrics on revealing the advantage and disadvantage of graph-aware modes on both synthetic and benchmark real-world datasets.

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ICML Conference 2022 Conference Paper

Generative Coarse-Graining of Molecular Conformations

  • Wujie Wang
  • Minkai Xu
  • Chen Cai
  • Benjamin Kurt Miller
  • Tess E. Smidt
  • Yusu Wang 0001
  • Jian Tang 0005
  • Rafael Gómez-Bombarelli

Coarse-graining (CG) of molecular simulations simplifies the particle representation by grouping selected atoms into pseudo-beads and therefore drastically accelerates simulation. However, such CG procedure induces information losses, which makes accurate backmapping, i. e. , restoring fine-grained (FG) coordinates from CG coordinates, a long-standing challenge. Inspired by the recent progress in generative models and equivariant networks, we propose a novel model that rigorously embeds the vital probabilistic nature and geometrical consistency requirements of the backmapping transformation. Our model encodes the FG uncertainties into an invariant latent space and decodes them back to FG geometries via equivariant convolutions. To standardize the evaluation of this domain, we further provide three comprehensive benchmarks based on molecular dynamics trajectories. Extensive experiments show that our approach always recovers more realistic structures and outperforms existing data-driven methods with a significant margin.

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ICLR Conference 2022 Conference Paper

GeoDiff: A Geometric Diffusion Model for Molecular Conformation Generation

  • Minkai Xu
  • Lantao Yu
  • Yang Song 0011
  • Chence Shi
  • Stefano Ermon
  • Jian Tang 0005

Predicting molecular conformations from molecular graphs is a fundamental problem in cheminformatics and drug discovery. Recently, significant progress has been achieved with machine learning approaches, especially with deep generative models. Inspired by the diffusion process in classical non-equilibrium thermodynamics where heated particles will diffuse from original states to a noise distribution, in this paper, we propose a novel generative model named GeoDiff for molecular conformation prediction. GeoDiff treats each atom as a particle and learns to directly reverse the diffusion process (i.e., transforming from a noise distribution to stable conformations) as a Markov chain. Modeling such a generation process is however very challenging as the likelihood of conformations should be roto-translational invariant. We theoretically show that Markov chains evolving with equivariant Markov kernels can induce an invariant distribution by design, and further propose building blocks for the Markov kernels to preserve the desirable equivariance property. The whole framework can be efficiently trained in an end-to-end fashion by optimizing a weighted variational lower bound to the (conditional) likelihood. Experiments on multiple benchmarks show that GeoDiff is superior or comparable to existing state-of-the-art approaches, especially on large molecules.

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ICML Conference 2021 Conference Paper

An End-to-End Framework for Molecular Conformation Generation via Bilevel Programming

  • Minkai Xu
  • Wujie Wang
  • Shitong Luo
  • Chence Shi
  • Yoshua Bengio
  • Rafael Gómez-Bombarelli
  • Jian Tang 0005

Predicting molecular conformations (or 3D structures) from molecular graphs is a fundamental problem in many applications. Most existing approaches are usually divided into two steps by first predicting the distances between atoms and then generating a 3D structure through optimizing a distance geometry problem. However, the distances predicted with such two-stage approaches may not be able to consistently preserve the geometry of local atomic neighborhoods, making the generated structures unsatisfying. In this paper, we propose an end-to-end solution for molecular conformation prediction called ConfVAE based on the conditional variational autoencoder framework. Specifically, the molecular graph is first encoded in a latent space, and then the 3D structures are generated by solving a principled bilevel optimization program. Extensive experiments on several benchmark data sets prove the effectiveness of our proposed approach over existing state-of-the-art approaches. Code is available at \url{https: //github. com/MinkaiXu/ConfVAE-ICML21}.

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AAMAS Conference 2021 Conference Paper

Energy-Based Imitation Learning

  • Minghuan Liu
  • Tairan He
  • Minkai Xu
  • Weinan Zhang

We tackle a common scenario in imitation learning (IL), where agents try to recover the optimal policy from expert demonstrations without further access to the expert or environment reward signals. Except the simple Behavior Cloning (BC) that adopts supervised learning followed by the problem of compounding error, previous solutions like inverse reinforcement learning (IRL) and recent generative adversarial methods involve a bi-level or alternating optimization for updating the reward function and the policy, suffering from high computational cost and training instability. Inspired by recent progress in energy-based model (EBM), in this paper, we propose a simplified IL framework named Energy-Based Imitation Learning (EBIL). Instead of updating the reward and policy iteratively, EBIL breaks out of the traditional IRL paradigm by a simple and flexible two-stage solution: first estimating the expert energy as the surrogate reward function through score matching, then utilizing such a reward for learning the policy by reinforcement learning algorithms. EBIL combines the idea of both EBM and occupancy measure matching, and via theoretic analysis we reveal that EBIL and Max-Entropy IRL (MaxEnt IRL) approaches are two sides of the same coin, and thus EBIL could be an alternative of adversarial IRL methods. Extensive experiments on qualitative and quantitative evaluations indicate that EBIL is able to recover meaningful and interpretative reward signals while achieving effective and comparable performance against existing algorithms on IL benchmarks.

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ICML Conference 2021 Conference Paper

Learning Gradient Fields for Molecular Conformation Generation

  • Chence Shi
  • Shitong Luo
  • Minkai Xu
  • Jian Tang 0005

We study a fundamental problem in computational chemistry known as molecular conformation generation, trying to predict stable 3D structures from 2D molecular graphs. Existing machine learning approaches usually first predict distances between atoms and then generate a 3D structure satisfying the distances, where noise in predicted distances may induce extra errors during 3D coordinate generation. Inspired by the traditional force field methods for molecular dynamics simulation, in this paper, we propose a novel approach called ConfGF by directly estimating the gradient fields of the log density of atomic coordinates. The estimated gradient fields allow directly generating stable conformations via Langevin dynamics. However, the problem is very challenging as the gradient fields are roto-translation equivariant. We notice that estimating the gradient fields of atomic coordinates can be translated to estimating the gradient fields of interatomic distances, and hence develop a novel algorithm based on recent score-based generative models to effectively estimate these gradients. Experimental results across multiple tasks show that ConfGF outperforms previous state-of-the-art baselines by a significant margin.

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ICLR Conference 2021 Conference Paper

Learning Neural Generative Dynamics for Molecular Conformation Generation

  • Minkai Xu
  • Shitong Luo
  • Yoshua Bengio
  • Jian Peng 0001
  • Jian Tang 0005

We study how to generate molecule conformations (i.e., 3D structures) from a molecular graph. Traditional methods, such as molecular dynamics, sample conformations via computationally expensive simulations. Recently, machine learning methods have shown great potential by training on a large collection of conformation data. Challenges arise from the limited model capacity for capturing complex distributions of conformations and the difficulty in modeling long-range dependencies between atoms. Inspired by the recent progress in deep generative models, in this paper, we propose a novel probabilistic framework to generate valid and diverse conformations given a molecular graph. We propose a method combining the advantages of both flow-based and energy-based models, enjoying: (1) a high model capacity to estimate the multimodal conformation distribution; (2) explicitly capturing the complex long-range dependencies between atoms in the observation space. Extensive experiments demonstrate the superior performance of the proposed method on several benchmarks, including conformation generation and distance modeling tasks, with a significant improvement over existing generative models for molecular conformation sampling.

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NeurIPS Conference 2021 Conference Paper

Predicting Molecular Conformation via Dynamic Graph Score Matching

  • Shitong Luo
  • Chence Shi
  • Minkai Xu
  • Jian Tang

Predicting stable 3D conformations from 2D molecular graphs has been a long-standing challenge in computational chemistry. Recently, machine learning approaches have demonstrated very promising results compared to traditional experimental and physics-based simulation methods. These approaches mainly focus on modeling the local interactions between neighboring atoms on the molecular graphs and overlook the long-range interactions between non-bonded atoms. However, these non-bonded atoms may be proximal to each other in 3D space, and modeling their interactions is of crucial importance to accurately determine molecular conformations, especially for large molecules and multi-molecular complexes. In this paper, we propose a new approach called Dynamic Graph Score Matching (DGSM) for molecular conformation prediction, which models both the local and long-range interactions by dynamically constructing graph structures between atoms according to their spatial proximity during both training and inference. Specifically, the DGSM directly estimates the gradient fields of the logarithm density of atomic coordinates according to the dynamically constructed graphs using score matching methods. The whole framework can be efficiently trained in an end-to-end fashion. Experiments across multiple tasks show that the DGSM outperforms state-of-the-art baselines by a large margin, and it is capable of generating conformations for a broader range of systems such as proteins and multi-molecular complexes.

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AAAI Conference 2021 Conference Paper

Towards Generalized Implementation of Wasserstein Distance in GANs

  • Minkai Xu

Wasserstein GANs (WGANs), built upon the Kantorovich- Rubinstein (KR) duality of Wasserstein distance, is one of the most theoretically sound GAN models. However, in practice it does not always outperform other variants of GANs. This is mostly due to the imperfect implementation of the Lipschitz condition required by the KR duality. Extensive work has been done in the community with different implementations of the Lipschitz constraint, which, however, is still hard to satisfy the restriction perfectly in practice. In this paper, we argue that the strong Lipschitz constraint might be unnecessary for optimization. Instead, we take a step back and try to relax the Lipschitz constraint. Theoretically, we first demonstrate a more general dual form of the Wasserstein distance called the Sobolev duality, which relaxes the Lipschitz constraint but still maintains the favorable gradient property of the Wasserstein distance. Moreover, we show that the KR duality is actually a special case of the Sobolev duality. Based on the relaxed duality, we further propose a generalized WGAN training scheme named Sobolev Wasserstein GAN, and empirically demonstrate the improvement over existing methods with extensive experiments.

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ICML Conference 2020 Conference Paper

A Graph to Graphs Framework for Retrosynthesis Prediction

  • Chence Shi
  • Minkai Xu
  • Hongyu Guo
  • Ming Zhang 0004
  • Jian Tang 0005

A fundamental problem in computational chemistry is to find a set of reactants to synthesize a target molecule, a. k. a. retrosynthesis prediction. Existing state-of-the-art methods rely on matching the target molecule with a large set of reaction templates, which are very computationally expensive and also suffer from the problem of coverage. In this paper, we propose a novel template-free approach called G2Gs by transforming a target molecular graph into a set of reactant molecular graphs. G2Gs first splits the target molecular graph into a set of synthons by identifying the reaction centers, and then translates the synthons to the final reactant graphs via a variational graph translation framework. Experimental results show that G2Gs significantly outperforms existing template-free approaches by up to 63% in terms of the top-1 accuracy and achieves a performance close to that of state-of-the-art template-based approaches, but does not require domain knowledge and is much more scalable.

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ICLR Conference 2020 Conference Paper

GraphAF: a Flow-based Autoregressive Model for Molecular Graph Generation

  • Chence Shi
  • Minkai Xu
  • Zhaocheng Zhu
  • Weinan Zhang 0001
  • Ming Zhang 0004
  • Jian Tang 0005

Molecular graph generation is a fundamental problem for drug discovery and has been attracting growing attention. The problem is challenging since it requires not only generating chemically valid molecular structures but also optimizing their chemical properties in the meantime. Inspired by the recent progress in deep generative models, in this paper we propose a flow-based autoregressive model for graph generation called GraphAF. GraphAF combines the advantages of both autoregressive and flow-based approaches and enjoys: (1) high model flexibility for data density estimation; (2) efficient parallel computation for training; (3) an iterative sampling process, which allows leveraging chemical domain knowledge for valency checking. Experimental results show that GraphAF is able to generate 68\% chemically valid molecules even without chemical knowledge rules and 100\% valid molecules with chemical rules. The training process of GraphAF is two times faster than the existing state-of-the-art approach GCPN. After fine-tuning the model for goal-directed property optimization with reinforcement learning, GraphAF achieves state-of-the-art performance on both chemical property optimization and constrained property optimization.

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AAAI Conference 2020 Conference Paper

Infomax Neural Joint Source-Channel Coding via Adversarial Bit Flip

  • Yuxuan Song
  • Minkai Xu
  • Lantao Yu
  • Hao Zhou
  • Shuo Shao
  • Yong Yu

Although Shannon theory states that it is asymptotically optimal to separate the source and channel coding as two independent processes, in many practical communication scenarios this decomposition is limited by the finite bit-length and computational power for decoding. Recently, neural joint source-channel coding (NECST) (Choi et al. 2018) is proposed to sidestep this problem. While it leverages the advancements of amortized inference and deep learning (Kingma and Welling 2013; Grover and Ermon 2018) to improve the encoding and decoding process, it still cannot always achieve compelling results in terms of compression and error correction performance due to the limited robustness of its learned coding networks. In this paper, motivated by the inherent connections between neural joint source-channel coding and discrete representation learning, we propose a novel regularization method called Infomax Adversarial-Bit- Flip (IABF) to improve the stability and robustness of the neural joint source-channel coding scheme. More specifically, on the encoder side, we propose to explicitly maximize the mutual information between the codeword and data; while on the decoder side, the amortized reconstruction is regularized within an adversarial framework. Extensive experiments conducted on various real-world datasets evidence that our IABF can achieve state-of-the-art performances on both compression and error correction benchmarks and outperform the baselines by a significant margin.

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