Machine learning methods, such as diffusion models, are widely explored as a promising way to accelerate high-fidelity fluid dynamics computation via a super-resolution process from faster-tocompute low-fidelity input. However, existing approaches usually make impractical assumptions that the low-fidelity data is downsampled from high-fidelity data. In reality, low-fidelity data is produced by numerical solvers that use a coarser resolution. Solvergenerated low-fidelity data usually sacrifices fine-grained details, such as small-scale vortices compared to high-fidelity ones. Our findings show that SOTA diffusion models struggle to reconstruct high-fidelity outputs from solver-generated low-fidelity inputs. To bridge this gap, we propose SG-Diff, a novel diffusion model for reconstruction, where both low-fidelity inputs and high-fidelity targets are generated from numerical solvers. We propose an Importance Weight strategy during training that serves as a form of self-guidance, focusing on intricate fluid details, and a Predictor-Corrector-Advancer SDE solver that embeds physical guidance into the diffusion sampling process. Together, these techniques steer the diffusion model toward more accurate reconstructions. Experimental results on four 2D turbulent flow datasets demonstrate the efficacy of SG-Diff against state-of-the-art baselines. Code, datasets, and additional appendix are available at https://github.com/RuoyanL i2002/Self-Guided-Diffusion-Model-for-Accelerating-Computationa l-Fluid-Dynamics.git

Molecular dynamics (MD) simulations are crucial for understanding and predicting the behavior of molecular systems in biology and chemistry. Yet, predicting long-term dynamics is still challenging. On one hand, it is hard to employ small-timestep models for long-term prediction, due to substantial rollout errors accumulated at each step, not to mention their extremely high time complexity due to the large number of rollout steps. On the other hand, it is hard to use large-timestep models to achieve high accuracy, due to their inability to capture subtle details of the dynamics. To bridge this dichotomy, we propose DoMiNO , a multi-scale framework that decomposes MD dynamics into several temporal resolutions, each governed by a neural graph ordinary differential equation (GraphODE) and is adaptively fused for final predictions. Concretely, DoMiNO operates through three key components: (1) an E(n)-equivariant graph neural network (EGNN) encoder that initializes latent states from a single observed molecular structure, maintaining SE(3) symmetries throughout; (2) a hierarchy of GraphODEs where each level captures scale-specific dynamics over normalized local time intervals, ranging from slow global motions to fast bond vibrations; and (3) an attention-based fusion module that adaptively combines multi-level predictions and reconstructs SE(3)-equivariant 3D coordinates. This design enables each hierarchical level to specialize in its characteristic timescale while preserving molecular symmetries. During inference, DoMiNO flexibly assembles predictions across different temporal resolutions, providing superior performance over both short-term and long-term dynamics. Empirical results on challenging MD benchmarks demonstrate that DoMiNO achieves significant improvements in prediction accuracy, particularly for molecules with pronounced timescale separation. The method exhibits significantly slower error growth over extended horizons compared to both single-scale baselines and state-of-the-art multi-step approaches. Our implementation is available at https://github.com/FrancoTSolis/DoMiNO-TKDD-Code .

Graph Neural Networks (GNNs) and differential equations (DEs) are two rapidly advancing areas of research that have shown remarkable synergy in recent years. GNNs have emerged as powerful tools for learning on graph-structured data, while differential equations provide a principled framework for modeling continuous dynamics across time and space. The intersection of these fields has led to innovative approaches that leverage the strengths of both, enabling applications in physics-informed learning, spatiotemporal modeling, and scientific computing. This survey aims to provide a comprehensive overview of the burgeoning research at the intersection of GNNs and DEs. We will categorize existing methods, discuss their underlying principles, and highlight their applications across domains such as molecular modeling, traffic prediction, and epidemic spreading. Furthermore, we identify open challenges and outline future research directions to advance this interdisciplinary field. A comprehensive paper list is provided at https://github.com/Emory-Melody/Awesome-Graph-NDEs.

Real-world multi-agent systems are often dynamic and continuous, where the agents co-evolve and undergo changes in their trajectories and interactions over time. For example, the COVID-19 transmission in the U.S. can be viewed as a multi-agent system, where states act as agents and daily population movements between them are interactions. Estimating the counterfactual outcomes in such systems enables accurate future predictions and effective decision-making, such as formulating COVID-19 policies. However, existing methods fail to model the continuous dynamic effects of treatments on the outcome, especially when multiple treatments (e.g., "stay-at-home" and "get-vaccine" policies) are applied simultaneously. To tackle this challenge, we propose Causal Graph Ordinary Differential Equations (CAG-ODE), a novel model that captures the continuous interaction among agents using a Graph Neural Network (GNN) as the ODE function. The key innovation of our model is to learn time-dependent representations of treatments and incorporate them into the ODE function, enabling precise predictions of potential outcomes. To mitigate confounding bias, we further propose two domain adversarial learning-based objectives, which enable our model to learn balanced continuous representations that are not affected by treatments or interference. Experiments on two datasets (i.e., COVID-19 and tumor growth) demonstrate the superior performance of our proposed model.

Multi-agent dynamical systems refer to scenarios where multiple units (aka agents) interact with each other and evolve collectively over time. For instance, people's health conditions are mutually influenced. Receiving vaccinations not only strengthens the long-term health status of one unit but also provides protection for those in their immediate surroundings. To make informed decisions in multi-agent dynamical systems, such as determining the optimal vaccine distribution plan, it is essential for decision-makers to estimate the continuous-time counterfactual outcomes. However, existing studies of causal inference over time rely on the assumption that units are mutually independent, which is not valid for multi-agent dynamical systems. In this paper, we aim to bridge this gap and study how to estimate counterfactual outcomes in multi-agent dynamical systems. Causal inference in a multi-agent dynamical system has unique challenges: 1) Confounders are time-varying and are present in both individual unit covariates and those of other units; 2) Units are affected by not only their own but also others' treatments; 3) The treatments are naturally dynamic, such as receiving vaccines and boosters in a seasonal manner. To this end, we model a multi-agent dynamical system as a graph and propose a novel model called CF-GODE (C ounterFactual Graph Ordinary Differential Equations). CF-GODE is a causal model that estimates continuous-time counterfactual outcomes in the presence of inter-dependencies between units. To facilitate continuous-time estimation, we propose Treatment-Induced GraphODE, a novel ordinary differential equation based on graph neural networks (GNNs), which can incorporate dynamical treatments as additional inputs to predict potential outcomes over time. To remove confounding bias, we propose two domain adversarial learning based objectives that learn balanced continuous representation trajectories, which are not predictive of treatments and interference. We further provide theoretical justification to prove their effectiveness. Experiments on two semi-synthetic datasets confirm that CF-GODE outperforms baselines on counterfactual estimation. We also provide extensive analyses to understand how our model works.

Learning multi-agent system dynamics have been extensively studied for various real-world applications, such as molecular dynamics in biology, multi-body system prediction in physics, and particle dynamics in material science. Most of the existing models are built to learn single system dynamics, which learn the dynamics from observed historical data and predict the future trajectory. In practice, however, we might observe multiple systems that are generated across different environments, which differ in latent exogenous factors such as temperature and gravity. One simple solution is to learn multiple environment-specific models, but it fails to exploit the potential commonalities among the dynamics across environments and offers poor prediction results where per-environment data is sparse or limited. Here, we present GG-ODE (Generalized Graph Ordinary Differential Equations), a machine learning framework for learning continuous multi-agent system dynamics across environments. Our model learns system dynamics using neural ordinary differential equations (ODE) parameterized by Graph Neural Networks (GNNs) to capture the continuous interaction among agents. We achieve the model generalization by assuming the dynamics across different environments are governed by common physics laws that can be captured via learning a shared ODE function. The distinct latent exogenous factors learned for each environment are incorporated into the ODE function to account for their differences. To improve model performance, we additionally design two regularization losses to (1) enforce the orthogonality between the learned initial states and exogenous factors via mutual information minimization; and (2) reduce the temporal variance of learned exogenous factors within the same system via contrastive learning. Experiments over various physical simulations show that our model can accurately predict system dynamics, especially in the long range, and can generalize well to new systems with few observations.

Many real-world systems such as social networks and moving planets are dynamic in nature, where a set of coupled objects are connected via the interaction graph and exhibit complex behavior along the time. For example, the COVID-19 pandemic can be considered as a dynamical system, where objects represent geographical locations (e.g., states) whose daily confirmed cases of infection evolve over time. Outbreak at one location may influence another location as people travel between these locations, forming a graph. Thus, how to model and predict the complex dynamics for these systems becomes a critical research problem. Existing work on modeling graph-structured data mostly assumes a static setting. How to handle dynamic graphs remains to be further explored. On one hand, features of objects change over time, influenced by the linked objects in the interaction graph. On the other hand, the graph itself can also evolve, where new interactions (links) may form and existing links may drop, which may in turn be affected by the dynamic features of objects. In this paper, we propose coupled graph ODE: a novel latent ordinary differential equation (ODE) generative model that learns the coupled dynamics of nodes and edges with a graph neural network (GNN) based ODE in a continuous manner. Our model consists of two coupled ODE functions for modeling the dynamics of edges and nodes based on their latent representations respectively. It employs a novel encoder parameterized by a GNN for inferring the initial states from historical data, which serves as the starting point of the predicted latent trajectories. Experiment results on the COVID-19 dataset and the simulated social network dataset demonstrate the effectiveness of our proposed method.

This paper describes a novel diffusion model, DyDiff-VAE, for information diffusion prediction on social media. Given the initial content and a sequence of forwarding users, DyDiff-VAE aims to estimate the propagation likelihood for other potential users and predict the corresponding user rankings. Inferring user interests from diffusion data lies the foundation of diffusion prediction, because users often forward the information in which they are interested or the information from those who share similar interests. Their interests also evolve over time as the result of the dynamic social influence from neighbors and the time-sensitive information gained inside/outside the social media. Existing works fail to model users' intrinsic interests from the diffusion data and assume user interests remain static along the time. DyDiff-VAE advances the state of the art in two directions: (i) We propose a dynamic encoder to infer the evolution of user interests from observed diffusion data. (ii) We propose a dual attentive decoder to estimate the propagation likelihood by integrating information from both the initial cascade content and the forwarding user sequence. Extensive experiments on four real-world datasets from Twitter and Youtube demonstrate the advantages of the proposed model; we show that it achieves 43.3%relative gains over the best baseline on average. Moreover, it has the lowest run-time compared with recurrent neural network based models.

Personalized hashtag recommendation for users could substantially promote user engagement in microblogging websites; users can discover microblogs aligned with their interests. However, user profiling on microblogging websites is challenging because most users tend not to generate content. Our core idea is to build a graph-based profile of users and incorporate it into hashtag recommendation. Indeed, user’s followee/follower links implicitly indicate their interests. Considering that microblogging networks are scale-free networks, to maintain the efficiency and effectiveness of the model, rather than analyzing the entire network, we model users based on their links towards hub nodes. That is, hashtags and hub nodes are projected into a shared latent space. To predict the relevance of a user to a hashtag, a projection of the user is built by aggregating the embeddings of her hub neighbors guided by an attention model and then compared with the hashtag. Classically, attention models can be trained in an end to end manner. However, due to the high complexity of our problem, we propose a novel weak supervision model for the attention component, which significantly improves the effectiveness of the model. We performed extensive experiments on two datasets collected from Twitter and Weibo, and the results confirm that our method substantially outperforms the baselines.