Zuobai Zhang

Minghao Xu, Zuobai Zhang, Jiarui Lu et al.

We are now witnessing significant progress of deep learning methods in a variety of tasks (or datasets) of proteins. However, there is a lack of a standard benchmark to evaluate the performance of different methods, which hinders the progress of deep learning in this field. In this paper, we propose such a benchmark called PEER, a comprehensive and multi-task benchmark for Protein sEquence undERstanding. PEER provides a set of diverse protein understanding tasks including protein function prediction, protein localization prediction, protein structure prediction, protein-protein interaction prediction, and protein-ligand interaction prediction. We evaluate different types of sequence-based methods for each task including traditional feature engineering approaches, different sequence encoding methods as well as large-scale pre-trained protein language models. In addition, we also investigate the performance of these methods under the multi-task learning setting. Experimental results show that large-scale pre-trained protein language models achieve the best performance for most individual tasks, and jointly training multiple tasks further boosts the performance. The datasets and source codes of this benchmark are all available at https://github.com/DeepGraphLearning/PEER_Benchmark

Yangtian Zhang, Zuobai Zhang, Bozitao Zhong et al.

Proteins play a critical role in carrying out biological functions, and their 3D structures are essential in determining their functions. Accurately predicting the conformation of protein side-chains given their backbones is important for applications in protein structure prediction, design and protein-protein interactions. Traditional methods are computationally intensive and have limited accuracy, while existing machine learning methods treat the problem as a regression task and overlook the restrictions imposed by the constant covalent bond lengths and angles. In this work, we present DiffPack, a torsional diffusion model that learns the joint distribution of side-chain torsional angles, the only degrees of freedom in side-chain packing, by diffusing and denoising on the torsional space. To avoid issues arising from simultaneous perturbation of all four torsional angles, we propose autoregressively generating the four torsional angles from $χ_1$ to $χ_4$ and training diffusion models for each torsional angle. We evaluate the method on several benchmarks for protein side-chain packing and show that our method achieves improvements of $11.9\%$ and $13.5\%$ in angle accuracy on CASP13 and CASP14, respectively, with a significantly smaller model size ($60\times$ fewer parameters). Additionally, we show the effectiveness of our method in enhancing side-chain predictions in the AlphaFold2 model. Code is available at https://github.com/DeepGraphLearning/DiffPack.

Zuobai Zhang, Minghao Xu, Arian Jamasb et al.

Learning effective protein representations is critical in a variety of tasks in biology such as predicting protein function or structure. Existing approaches usually pretrain protein language models on a large number of unlabeled amino acid sequences and then finetune the models with some labeled data in downstream tasks. Despite the effectiveness of sequence-based approaches, the power of pretraining on known protein structures, which are available in smaller numbers only, has not been explored for protein property prediction, though protein structures are known to be determinants of protein function. In this paper, we propose to pretrain protein representations according to their 3D structures. We first present a simple yet effective encoder to learn the geometric features of a protein. We pretrain the protein graph encoder by leveraging multiview contrastive learning and different self-prediction tasks. Experimental results on both function prediction and fold classification tasks show that our proposed pretraining methods outperform or are on par with the state-of-the-art sequence-based methods, while using much less pretraining data. Our implementation is available at https://github.com/DeepGraphLearning/GearNet.

Zuobai Zhang, Chuanrui Wang, Minghao Xu et al.

Learning effective protein representations is critical in a variety of tasks in biology such as predicting protein functions. Recent sequence representation learning methods based on Protein Language Models (PLMs) excel in sequence-based tasks, but their direct adaptation to tasks involving protein structures remains a challenge. In contrast, structure-based methods leverage 3D structural information with graph neural networks and geometric pre-training methods show potential in function prediction tasks, but still suffers from the limited number of available structures. To bridge this gap, our study undertakes a comprehensive exploration of joint protein representation learning by integrating a state-of-the-art PLM (ESM-2) with distinct structure encoders (GVP, GearNet, CDConv). We introduce three representation fusion strategies and explore different pre-training techniques. Our method achieves significant improvements over existing sequence- and structure-based methods, setting new state-of-the-art for function annotation. This study underscores several important design choices for fusing protein sequence and structure information. Our implementation is available at https://github.com/DeepGraphLearning/ESM-GearNet.

Zhaocheng Zhu, Mikhail Galkin, Zuobai Zhang et al. · deepmind

Answering complex first-order logic (FOL) queries on knowledge graphs is a fundamental task for multi-hop reasoning. Traditional symbolic methods traverse a complete knowledge graph to extract the answers, which provides good interpretation for each step. Recent neural methods learn geometric embeddings for complex queries. These methods can generalize to incomplete knowledge graphs, but their reasoning process is hard to interpret. In this paper, we propose Graph Neural Network Query Executor (GNN-QE), a neural-symbolic model that enjoys the advantages of both worlds. GNN-QE decomposes a complex FOL query into relation projections and logical operations over fuzzy sets, which provides interpretability for intermediate variables. To reason about the missing links, GNN-QE adapts a graph neural network from knowledge graph completion to execute the relation projections, and models the logical operations with product fuzzy logic. Experiments on 3 datasets show that GNN-QE significantly improves over previous state-of-the-art models in answering FOL queries. Meanwhile, GNN-QE can predict the number of answers without explicit supervision, and provide visualizations for intermediate variables.

Sha Yuan, Hanyu Zhao, Shuai Zhao et al. · bytedance, pku

With the rapid development of deep learning, training Big Models (BMs) for multiple downstream tasks becomes a popular paradigm. Researchers have achieved various outcomes in the construction of BMs and the BM application in many fields. At present, there is a lack of research work that sorts out the overall progress of BMs and guides the follow-up research. In this paper, we cover not only the BM technologies themselves but also the prerequisites for BM training and applications with BMs, dividing the BM review into four parts: Resource, Models, Key Technologies and Application. We introduce 16 specific BM-related topics in those four parts, they are Data, Knowledge, Computing System, Parallel Training System, Language Model, Vision Model, Multi-modal Model, Theory&Interpretability, Commonsense Reasoning, Reliability&Security, Governance, Evaluation, Machine Translation, Text Generation, Dialogue and Protein Research. In each topic, we summarize clearly the current studies and propose some future research directions. At the end of this paper, we conclude the further development of BMs in a more general view.

Zuobai Zhang, Minghao Xu, Aurélie Lozano et al.

Self-supervised pre-training methods on proteins have recently gained attention, with most approaches focusing on either protein sequences or structures, neglecting the exploration of their joint distribution, which is crucial for a comprehensive understanding of protein functions by integrating co-evolutionary information and structural characteristics. In this work, inspired by the success of denoising diffusion models in generative tasks, we propose the DiffPreT approach to pre-train a protein encoder by sequence-structure joint diffusion modeling. DiffPreT guides the encoder to recover the native protein sequences and structures from the perturbed ones along the joint diffusion trajectory, which acquires the joint distribution of sequences and structures. Considering the essential protein conformational variations, we enhance DiffPreT by a method called Siamese Diffusion Trajectory Prediction (SiamDiff) to capture the correlation between different conformers of a protein. SiamDiff attains this goal by maximizing the mutual information between representations of diffusion trajectories of structurally-correlated conformers. We study the effectiveness of DiffPreT and SiamDiff on both atom- and residue-level structure-based protein understanding tasks. Experimental results show that the performance of DiffPreT is consistently competitive on all tasks, and SiamDiff achieves new state-of-the-art performance, considering the mean ranks on all tasks. Our implementation is available at https://github.com/DeepGraphLearning/SiamDiff.

Songtao Liu, Zhengkai Tu, Minkai Xu et al.

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.

Jiarui Lu, Bozitao Zhong, Zuobai Zhang et al.

The dynamic nature of proteins is crucial for determining their biological functions and properties, for which Monte Carlo (MC) and molecular dynamics (MD) simulations stand as predominant tools to study such phenomena. By utilizing empirically derived force fields, MC or MD simulations explore the conformational space through numerically evolving the system via Markov chain or Newtonian mechanics. However, the high-energy barrier of the force fields can hamper the exploration of both methods by the rare event, resulting in inadequately sampled ensemble without exhaustive running. Existing learning-based approaches perform direct sampling yet heavily rely on target-specific simulation data for training, which suffers from high data acquisition cost and poor generalizability. Inspired by simulated annealing, we propose Str2Str, a novel structure-to-structure translation framework capable of zero-shot conformation sampling with roto-translation equivariant property. Our method leverages an amortized denoising score matching objective trained on general crystal structures and has no reliance on simulation data during both training and inference. Experimental results across several benchmarking protein systems demonstrate that Str2Str outperforms previous state-of-the-art generative structure prediction models and can be orders of magnitude faster compared to long MD simulations. Our open-source implementation is available at https://github.com/lujiarui/Str2Str

Chuanrui Wang, Bozitao Zhong, Zuobai Zhang et al.

Structure-based protein design has attracted increasing interest, with numerous methods being introduced in recent years. However, a universally accepted method for evaluation has not been established, since the wet-lab validation can be overly time-consuming for the development of new algorithms, and the $\textit{in silico}$ validation with recovery and perplexity metrics is efficient but may not precisely reflect true foldability. To address this gap, we introduce two novel metrics: refoldability-based metric, which leverages high-accuracy protein structure prediction models as a proxy for wet lab experiments, and stability-based metric, which assesses whether models can assign high likelihoods to experimentally stable proteins. We curate datasets from high-quality CATH protein data, high-throughput $\textit{de novo}$ designed proteins, and mega-scale experimental mutagenesis experiments, and in doing so, present the $\textbf{PDB-Struct}$ benchmark that evaluates both recent and previously uncompared protein design methods. Experimental results indicate that ByProt, ProteinMPNN, and ESM-IF perform exceptionally well on our benchmark, while ESM-Design and AF-Design fall short on the refoldability metric. We also show that while some methods exhibit high sequence recovery, they do not perform as well on our new benchmark. Our proposed benchmark paves the way for a fair and comprehensive evaluation of protein design methods in the future. Code is available at https://github.com/WANG-CR/PDB-Struct.

Kieran Didi, Zuobai Zhang, Guoqing Zhou et al.

Protein interaction modeling is central to protein design, which has been transformed by machine learning with applications in drug discovery and beyond. In this landscape, structure-based de novo binder design is cast as either conditional generative modeling or sequence optimization via structure predictors ("hallucination"). We argue that this is a false dichotomy and propose Proteina-Complexa, a novel fully atomistic binder generation method unifying both paradigms. We extend recent flow-based latent protein generation architectures and leverage the domain-domain interactions of monomeric computationally predicted protein structures to construct Teddymer, a new large-scale dataset of synthetic binder-target pairs for pretraining. Combined with high-quality experimental multimers, this enables training a strong base model. We then perform inference-time optimization with this generative prior, unifying the strengths of previously distinct generative and hallucination methods. Proteina-Complexa sets a new state of the art in computational binder design benchmarks: it delivers markedly higher in-silico success rates than existing generative approaches, and our novel test-time optimization strategies greatly outperform previous hallucination methods under normalized compute budgets. We also demonstrate interface hydrogen bond optimization, fold class-guided binder generation, and extensions to small molecule targets and enzyme design tasks, again surpassing prior methods. Code, models and new data will be publicly released.

Xinyu Yuan, Xixian Liu, Ya Shi Zhang et al.

Building Virtual Cells that can accurately simulate cellular responses to perturbations is a long-standing goal in systems biology. A fundamental challenge is that high-throughput single-cell sequencing is destructive: the same cell cannot be observed both before and after a perturbation. Thus, perturbation prediction requires mapping unpaired control and perturbed populations. Existing models address this by learning maps between distributions, but typically assume a single fixed response distribution when conditioned on observed cellular context (e.g., cell type) and the perturbation type. In reality, responses vary systematically due to unobservable latent factors such as microenvironmental fluctuations and complex batch effects, forming a manifold of possible distributions for the same observed conditions. To account for this variability, we introduce PerturbDiff, which shifts modeling from individual cells to entire distributions. By embedding distributions as points in a Hilbert space, we define a diffusion-based generative process operating directly over probability distributions. This allows PerturbDiff to capture population-level response shifts across hidden factors. Benchmarks on established datasets show that PerturbDiff achieves state-of-the-art performance in single-cell response prediction and generalizes substantially better to unseen perturbations. See our project page (https://katarinayuan.github.io/PerturbDiff-ProjectPage/), where code and data will be made publicly available (https://github.com/DeepGraphLearning/PerturbDiff).

Xinyu Yuan, Zhihao Zhan, Zuobai Zhang et al.

Transcriptome foundation models TFMs hold great promises of deciphering the transcriptomic language that dictate diverse cell functions by self-supervised learning on large-scale single-cell gene expression data, and ultimately unraveling the complex mechanisms of human diseases. However, current TFMs treat cells as independent samples and ignore the taxonomic relationships between cell types, which are available in cell ontology graphs. We argue that effectively leveraging this ontology information during the TFM pre-training can improve learning biologically meaningful gene co-expression patterns while preserving TFM as a general purpose foundation model for downstream zero-shot and fine-tuning tasks. To this end, we present single cell, Cell-ontology guided TFM scCello. We introduce cell-type coherence loss and ontology alignment loss, which are minimized along with the masked gene expression prediction loss during the pre-training. The novel loss component guide scCello to learn the cell-type-specific representation and the structural relation between cell types from the cell ontology graph, respectively. We pre-trained scCello on 22 million cells from CellxGene database leveraging their cell-type labels mapped to the cell ontology graph from Open Biological and Biomedical Ontology Foundry. Our TFM demonstrates competitive generalization and transferability performance over the existing TFMs on biologically important tasks including identifying novel cell types of unseen cells, prediction of cell-type-specific marker genes, and cancer drug responses.

Zuobai Zhang, Jiarui Lu, Vijil Chenthamarakshan et al.

Protein language models are a powerful tool for learning protein representations through pre-training on vast protein sequence datasets. However, traditional protein language models lack explicit structural supervision, despite its relevance to protein function. To address this issue, we introduce the integration of remote homology detection to distill structural information into protein language models without requiring explicit protein structures as input. We evaluate the impact of this structure-informed training on downstream protein function prediction tasks. Experimental results reveal consistent improvements in function annotation accuracy for EC number and GO term prediction. Performance on mutant datasets, however, varies based on the relationship between targeted properties and protein structures. This underscores the importance of considering this relationship when applying structure-aware training to protein function prediction tasks. Code and model weights are available at https://github.com/DeepGraphLearning/esm-s.

Wanyue Xu, Yubo Sun, Mingzhe Zhu et al.

The phenomenon of opinion disagreement has been empirically observed and reported in the literature, which is affected by various factors, such as the structure of social networks. An important discovery in network science is that most real-life networks, including social networks, are scale-free and sparse. In this paper, we study noisy opinion dynamics in sparse scale-free social networks to uncover the influence of power-law topology on opinion disagreement. We adopt the popular discrete-time DeGroot model for opinion dynamics in a graph, where nodes' opinions are subject to white noise. We first study opinion disagreement in many realistic and model networks with a scale-free topology, which approaches a constant, indicating that a scale-free structure is resistant to noise in the opinion dynamics. Moreover, existing algorithms for estimating opinion disagreement are computationally impractical for large-scale networks due to their high computational complexity. To solve this challenge, we introduce a sublinear-time algorithm to approximate this quantity with a theoretically guaranteed error. This algorithm efficiently simulates truncated random walks starting from a subset of nodes while preserving accurate estimation. Extensive experiments demonstrate its efficiency, accuracy, and scalability.

Zuobai Zhang, Pascal Notin, Yining Huang et al.

Designing novel functional proteins crucially depends on accurately modeling their fitness landscape. Given the limited availability of functional annotations from wet-lab experiments, previous methods have primarily relied on self-supervised models trained on vast, unlabeled protein sequence or structure datasets. While initial protein representation learning studies solely focused on either sequence or structural features, recent hybrid architectures have sought to merge these modalities to harness their respective strengths. However, these sequence-structure models have so far achieved only incremental improvements when compared to the leading sequence-only approaches, highlighting unresolved challenges effectively leveraging these modalities together. Moreover, the function of certain proteins is highly dependent on the granular aspects of their surface topology, which have been overlooked by prior models. To address these limitations, we introduce the Sequence-Structure-Surface Fitness (S3F) model - a novel multimodal representation learning framework that integrates protein features across several scales. Our approach combines sequence representations from a protein language model with Geometric Vector Perceptron networks encoding protein backbone and detailed surface topology. The proposed method achieves state-of-the-art fitness prediction on the ProteinGym benchmark encompassing 217 substitution deep mutational scanning assays, and provides insights into the determinants of protein function. Our code is at https://github.com/DeepGraphLearning/S3F.

Arian R. Jamasb, Alex Morehead, Chaitanya K. Joshi et al.

We introduce ProteinWorkshop, a comprehensive benchmark suite for representation learning on protein structures with Geometric Graph Neural Networks. We consider large-scale pre-training and downstream tasks on both experimental and predicted structures to enable the systematic evaluation of the quality of the learned structural representation and their usefulness in capturing functional relationships for downstream tasks. We find that: (1) large-scale pretraining on AlphaFold structures and auxiliary tasks consistently improve the performance of both rotation-invariant and equivariant GNNs, and (2) more expressive equivariant GNNs benefit from pretraining to a greater extent compared to invariant models. We aim to establish a common ground for the machine learning and computational biology communities to rigorously compare and advance protein structure representation learning. Our open-source codebase reduces the barrier to entry for working with large protein structure datasets by providing: (1) storage-efficient dataloaders for large-scale structural databases including AlphaFoldDB and ESM Atlas, as well as (2) utilities for constructing new tasks from the entire PDB. ProteinWorkshop is available at: github.com/a-r-j/ProteinWorkshop.

Shengchao Liu, Meng Qu, Zuobai Zhang et al.

Multi-task learning for molecular property prediction is becoming increasingly important in drug discovery. However, in contrast to other domains, the performance of multi-task learning in drug discovery is still not satisfying as the number of labeled data for each task is too limited, which calls for additional data to complement the data scarcity. In this paper, we study multi-task learning for molecular property prediction in a novel setting, where a relation graph between tasks is available. We first construct a dataset (ChEMBL-STRING) including around 400 tasks as well as a task relation graph. Then to better utilize such relation graph, we propose a method called SGNN-EBM to systematically investigate the structured task modeling from two perspectives. (1) In the \emph{latent} space, we model the task representations by applying a state graph neural network (SGNN) on the relation graph. (2) In the \emph{output} space, we employ structured prediction with the energy-based model (EBM), which can be efficiently trained through noise-contrastive estimation (NCE) approach. Empirical results justify the effectiveness of SGNN-EBM. Code is available on https://github.com/chao1224/SGNN-EBM.

Tomas Geffner, Kieran Didi, Zuobai Zhang et al.

Recently, diffusion- and flow-based generative models of protein structures have emerged as a powerful tool for de novo protein design. Here, we develop Proteina, a new large-scale flow-based protein backbone generator that utilizes hierarchical fold class labels for conditioning and relies on a tailored scalable transformer architecture with up to 5x as many parameters as previous models. To meaningfully quantify performance, we introduce a new set of metrics that directly measure the distributional similarity of generated proteins with reference sets, complementing existing metrics. We further explore scaling training data to millions of synthetic protein structures and explore improved training and sampling recipes adapted to protein backbone generation. This includes fine-tuning strategies like LoRA for protein backbones, new guidance methods like classifier-free guidance and autoguidance for protein backbones, and new adjusted training objectives. Proteina achieves state-of-the-art performance on de novo protein backbone design and produces diverse and designable proteins at unprecedented length, up to 800 residues. The hierarchical conditioning offers novel control, enabling high-level secondary-structure guidance as well as low-level fold-specific generation.

Tomas Geffner, Kieran Didi, Zhonglin Cao et al.

Recently, many generative models for de novo protein structure design have emerged. Yet, only few tackle the difficult task of directly generating fully atomistic structures jointly with the underlying amino acid sequence. This is challenging, for instance, because the model must reason over side chains that change in length during generation. We introduce La-Proteina for atomistic protein design based on a novel partially latent protein representation: coarse backbone structure is modeled explicitly, while sequence and atomistic details are captured via per-residue latent variables of fixed dimensionality, thereby effectively side-stepping challenges of explicit side-chain representations. Flow matching in this partially latent space then models the joint distribution over sequences and full-atom structures. La-Proteina achieves state-of-the-art performance on multiple generation benchmarks, including all-atom co-designability, diversity, and structural validity, as confirmed through detailed structural analyses and evaluations. Notably, La-Proteina also surpasses previous models in atomistic motif scaffolding performance, unlocking critical atomistic structure-conditioned protein design tasks. Moreover, La-Proteina is able to generate co-designable proteins of up to 800 residues, a regime where most baselines collapse and fail to produce valid samples, demonstrating La-Proteina's scalability and robustness.

Jiarui Lu, Zuobai Zhang, Bozitao Zhong et al.

The protein dynamics are common and important for their biological functions and properties, the study of which usually involves time-consuming molecular dynamics (MD) simulations in silico. Recently, generative models has been leveraged as a surrogate sampler to obtain conformation ensembles with orders of magnitude faster and without requiring any simulation data (a "zero-shot" inference). However, being agnostic of the underlying energy landscape, the accuracy of such generative model may still be limited. In this work, we explore the few-shot setting of such pre-trained generative sampler which incorporates MD simulations in a tractable manner. Specifically, given a target protein of interest, we first acquire some seeding conformations from the pre-trained sampler followed by a number of physical simulations in parallel starting from these seeding samples. Then we fine-tuned the generative model using the simulation trajectories above to become a target-specific sampler. Experimental results demonstrated the superior performance of such few-shot conformation sampler at a tractable computational cost.

Zuobai Zhang, Jiarui Lu, Vijil Chenthamarakshan et al.

Protein function annotation is an important yet challenging task in biology. Recent deep learning advancements show significant potential for accurate function prediction by learning from protein sequences and structures. Nevertheless, these predictor-based methods often overlook the modeling of protein similarity, an idea commonly employed in traditional approaches using sequence or structure retrieval tools. To fill this gap, we first study the effect of inter-protein similarity modeling by benchmarking retriever-based methods against predictors on protein function annotation tasks. Our results show that retrievers can match or outperform predictors without large-scale pre-training. Building on these insights, we introduce a novel variational pseudo-likelihood framework, ProtIR, designed to improve function predictors by incorporating inter-protein similarity modeling. This framework iteratively refines knowledge between a function predictor and retriever, thereby combining the strengths of both predictors and retrievers. ProtIR showcases around 10% improvement over vanilla predictor-based methods. Besides, it achieves performance on par with protein language model-based methods, yet without the need for massive pre-training, highlighting the efficacy of our framework. Code will be released upon acceptance.

Zhaocheng Zhu, Chence Shi, Zuobai Zhang et al.

Machine learning has huge potential to revolutionize the field of drug discovery and is attracting increasing attention in recent years. However, lacking domain knowledge (e.g., which tasks to work on), standard benchmarks and data preprocessing pipelines are the main obstacles for machine learning researchers to work in this domain. To facilitate the progress of machine learning for drug discovery, we develop TorchDrug, a powerful and flexible machine learning platform for drug discovery built on top of PyTorch. TorchDrug benchmarks a variety of important tasks in drug discovery, including molecular property prediction, pretrained molecular representations, de novo molecular design and optimization, retrosynthsis prediction, and biomedical knowledge graph reasoning. State-of-the-art techniques based on geometric deep learning (or graph machine learning), deep generative models, reinforcement learning and knowledge graph reasoning are implemented for these tasks. TorchDrug features a hierarchical interface that facilitates customization from both novices and experts in this domain. Tutorials, benchmark results and documentation are available at https://torchdrug.ai. Code is released under Apache License 2.0.

Zhaocheng Zhu, Zuobai Zhang, Louis-Pascal Xhonneux et al.

Link prediction is a very fundamental task on graphs. Inspired by traditional path-based methods, in this paper we propose a general and flexible representation learning framework based on paths for link prediction. Specifically, we define the representation of a pair of nodes as the generalized sum of all path representations, with each path representation as the generalized product of the edge representations in the path. Motivated by the Bellman-Ford algorithm for solving the shortest path problem, we show that the proposed path formulation can be efficiently solved by the generalized Bellman-Ford algorithm. To further improve the capacity of the path formulation, we propose the Neural Bellman-Ford Network (NBFNet), a general graph neural network framework that solves the path formulation with learned operators in the generalized Bellman-Ford algorithm. The NBFNet parameterizes the generalized Bellman-Ford algorithm with 3 neural components, namely INDICATOR, MESSAGE and AGGREGATE functions, which corresponds to the boundary condition, multiplication operator, and summation operator respectively. The NBFNet is very general, covers many traditional path-based methods, and can be applied to both homogeneous graphs and multi-relational graphs (e.g., knowledge graphs) in both transductive and inductive settings. Experiments on both homogeneous graphs and knowledge graphs show that the proposed NBFNet outperforms existing methods by a large margin in both transductive and inductive settings, achieving new state-of-the-art results.