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 .

Learning curve extrapolation predicts neural network performance from early training epochs and has been applied to accelerate AutoML, facilitating hyperparameter tuning and neural architecture search. However, existing methods typically model the evolution of learning curves in isolation, neglecting the impact of neural network (NN) architectures, which influence the loss landscape and learning trajectories. In this work, we explore whether incorporating neural network architecture improves learning curve modeling and how to effectively integrate this architectural information. Motivated by the dynamical system view of optimization, we propose a novel architecture-aware neural differential equation model to forecast learning curves continuously. We empirically demonstrate its ability to capture the general trend of fluctuating learning curves while quantifying uncertainty through variational parameters. Our model outperforms current state-of-the-art learning curve extrapolation methods and pure time-series modeling approaches for both MLP and CNN-based learning curves. Additionally, we explore the applicability of our method in Neural Architecture Search scenarios, such as training configuration ranking.

Generative Language Models rely on autoregressive decoding to produce the output sequence token by token.Many tasks such as preference optimization, require the model to produce task-level output consisting of multiple tokens directly by selecting candidates from a pool as predictions.Determining a task-level prediction from candidates using the ordinary token-level decoding mechanism is constrained by time-consuming decoding and interrupted gradients by discrete token selection.Existing works have been using decoding-free candidate selection methods to obtain candidate probability from initial output logits over vocabulary.Though these estimation methods are widely used, they are not systematically evaluated, especially on end tasks.We introduce an evaluation of a comprehensive collection of decoding-free candidate selection approaches on a comprehensive set of tasks, including five multiple-choice QA tasks with a small candidate pool and four clinical decision tasks with a massive amount of candidates, some with 10k+ options.We evaluate the estimation methods paired with a wide spectrum of foundation LMs covering different architectures, sizes and training paradigms.The results and insights from our analysis inform the future model design.

Neural Ordinary Differential Equations (Neural ODEs or NODEs) excel at modeling continuous dynamical systems from observational data, especially when the data is irregularly sampled. However, existing training methods predominantly rely on numerical ODE solvers, which are time-consuming and prone to accumulating numerical errors over time due to autoregression. In this work, we propose VF-NODE, a novel approach based on the variational formulation (VF) to accelerate the training of NODEs. Unlike existing training methods, the proposed VF-NODEs implement a series of global integrals, thus evaluating Deep Neural Network (DNN)--based vector fields only at specific observed data points. This strategy drastically reduces the number of function evaluations (NFEs). Moreover, our method eliminates the use of autoregression, thereby reducing error accumulations for modeling dynamical systems. Nevertheless, the VF loss introduces oscillatory terms into the integrals when using the Fourier basis. We incorporate Filon's method to address this issue. To further enhance the performance for noisy and incomplete data, we employ the natural cubic spline regression to estimate a closed-form approximation. We provide a fundamental analysis of how our approach minimizes computational costs. Extensive experiments demonstrate that our approach accelerates NODE training by 10 to 1000 times compared to existing NODE-based methods, while achieving higher or comparable accuracy in dynamical systems. The code is available at https://github.com/ZhaoHongjue/VF-NODE-ICLR2025.

Coupled dynamical systems govern essential phenomena across physics, biology, and engineering, where components interact through complex dependencies. While Graph Ordinary Differential Equations (GraphODE) offer a powerful framework to model these systems, their generalization capabilities degrade severely under limited observational training data due to two fundamental flaws: (i) the entanglement of static attributes and dynamic states in the initialization process, and (ii) the reliance on context-specific coupling patterns during training, which hinders performance in unseen scenarios. In this paper, we propose a Generalizable GraphODE with disentanglement and regularization (GREAT) to address these challenges. Through systematic analysis via the Structural Causal Model, we identify backdoor paths that undermine generalization and design two key modules to mitigate their effects. The Dynamic-Static Equilibrium Decoupler (DyStaED) disentangles static and dynamic states via orthogonal subspace projections, ensuring robust initialization. Furthermore, the Causal Mediation for Coupled Dynamics (CMCD) employs variational inference to estimate latent causal factors, reducing spurious correlations and enhancing universal coupling dynamics. Extensive experiments across diverse dynamical systems demonstrate that ours outperforms state-of-the-art methods within both in-distribution and out-of-distribution.

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.

Flow‐field reconstruction from sparse sensor measurements remains a central challenge in modern fluid dynamics, as the need for high‐fidelity data often conflicts with practical limits on sensor deployment. Existing deep learning–based methods have demonstrated promising results, but they typically depend on simplifying assumptions such as two‐dimensional domains, predefined governing equations, synthetic datasets derived from idealized flow physics, and unconstrained sensor placement. In this work, we address these limitations by studying flow reconstruction under realistic conditions and introducing a \emph{directional transport‐aware Graph Neural Network (GNN)} that explicitly encodes both flow directionality and information transport. We further show that conventional sensor placement strategies frequently yield suboptimal configurations. To overcome this, we propose a novel \emph{Two‐Step Constrained PPO} procedure for Proximal Policy Optimization (PPO), which jointly optimizes sensor layouts by incorporating flow variability and accounts for reconstruction model's performance disparity with respect to sensor placement. We conduct comprehensive experiments under realistic assumptions to benchmark the performance of our reconstruction model and sensor placement policy. Together, they achieve significant improvements over existing methods.

Recent studies reveal that large language models (LLMs) often struggle to resolve conflicting instructions embedded within hierarchical prompts, resulting in decreased compliance with system-level directives and compromising the reliability of safety-critical applications. While earlier approaches attempt to improve instruction hierarchy awareness through prompt engineering or embedding-level modifications, they typically lack structural modeling and either offer limited gains or require extensive fine-tuning. In this work, we introduce $\textbf{FocalLoRA}$, a parameter-efficient and structure-aware framework that strengthens hierarchical instruction adherence by selectively optimizing structurally critical attention heads, referred to as $\textit{focal heads}$, which exhibit heightened sensitivity to instruction conflicts. Experiments across multiple models and a dedicated benchmark demonstrate that FocalLoRA markedly enhances system instruction compliance with minimal tuning cost. For instance, on Llama-8B, fine-tuning only 0.0188\% of parameters yields a 35.52\% $\uparrow$ in system instruction compliance.

Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating physics-informed priors, such as in Hamiltonian Neural Networks (HNNs), achieves high-precision modeling for energy-conservative systems. However, real-world systems often deviate from strict energy conservation and follow different physical priors. To address this, we present a framework that achieves high-precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via a novel regularization term. It helps preserve energies for conservative systems while serving as a strong inductive bias for non-conservative, reversible systems. While TRS is a domain-specific physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation models, the proposed model TREAT demonstrates superior performance on diverse physical systems. It achieves a significant 11.5% MSE improvement in a challenging chaotic triple-pendulum scenario, underscoring TREAT’s broad applicability and effectiveness.

The rapid evolution of large language models (LLMs) has expanded their capabilities across various data modalities, extending from well-established image data to increasingly popular graph data. Given the limitation of LLMs in hallucinations and inaccuracies in recalling factual knowledge, Knowledge Graph (KG) has emerged as a crucial data modality to support more accurate reasoning by LLMs. However, integrating structured knowledge from KGs into LLMs remains challenging, as most current KG-enhanced LLM methods directly convert the KG into linearized text triples, which is not as expressive as the original structured data. To address this, we introduce GraphVis, which conserves the intricate graph structure through the visual modality to enhance the comprehension of KGs with the aid of Large Vision Language Models (LVLMs). Our approach incorporates a unique curriculum fine-tuning scheme which first instructs LVLMs to recognize basic graphical features from the images, and subsequently incorporates reasoning on QA tasks with the visual graphs. This cross-modal methodology not only markedly enhances performance on standard textual QA but also shows improved zero-shot VQA performance by utilizing synthetic graph images to augment the data for VQA tasks. We present comprehensive evaluations across commonsense reasoning QA benchmarks, where GraphVis provides an average improvement of 11.1% over its base model and outperforms existing KG-enhanced LLM approaches. Across VQA benchmarks such as ScienceQA that share similar scientific diagram images, GraphVis provides a notable gain of 4.32%.

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.

Leading graph ordinary differential equation (ODE) models have offered generalized strategies to model interacting multi-agent dynamical systems in a data-driven approach. They typically consist of a temporal graph encoder to get the initial states and a neural ODE-based generative model to model the evolution of dynamical systems. However, existing methods have severe deficiencies in capacity and efficiency due to the failure to model high-order correlations in long-term temporal trends. To tackle this, in this paper, we propose a novel model named High-order graph ODE (HOPE) for learning from dynamic interaction data, which can be naturally represented as a graph. It first adopts a twin graph encoder to initialize the latent state representations of nodes and edges, which consists of two branches to capture spatio-temporal correlations in complementary manners. More importantly, our HOPE utilizes a second-order graph ODE function which models the dynamics for both nodes and edges in the latent space respectively, which enables efficient learning of long-term dependencies from complex dynamical systems. Experiment results on a variety of datasets demonstrate both the effectiveness and efficiency of our proposed method.

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.

Modeling interacting dynamical systems, such as fluid dynamics and intermolecular interactions, is a fundamental research problem for understanding and simulating complex real-world systems. Many of these systems can be naturally represented by dynamic graphs, and graph neural network-based approaches have been proposed and shown promising performance. However, most of these approaches assume the underlying dynamics does not change over time, which is unfortunately untrue. For example, a molecular dynamics can be affected by the environment temperature over the time. In this paper, we take an attempt to provide a probabilistic view for time-varying dynamics and propose a model Context-attended Graph ODE (CARE) for modeling time-varying interacting dynamical systems. In our CARE, we explicitly use a context variable to model time-varying environment and construct an encoder to initialize the context variable from historical trajectories. Furthermore, we employ a neural ODE model to depict the dynamic evolution of the context variable inferred from system states. This context variable is incorporated into a coupled ODE to simultaneously drive the evolution of systems. Comprehensive experiments on four datasets demonstrate the effectiveness of our proposed CARE compared with several state-of-the-art approaches.

Zijie Huang, Zheng Li, Haoming Jiang, Tianyu Cao, Hanqing Lu, Bing Yin, Karthik Subbian, Yizhou Sun, Wei Wang. Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2022.

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.

Many real-world systems, such as moving planets, can be considered as multi-agent dynamic systems, where objects interact with each other and co-evolve along with the time. Such dynamics is usually difficult to capture, and understanding and predicting the dynamics based on observed trajectories of objects become a critical research problem in many domains. Most existing algorithms, however, assume the observations are regularly sampled and all the objects can be fully observed at each sampling time, which is impractical for many applications. In this paper, we pro-pose to learn system dynamics from irregularly-sampled and partial observations with underlying graph structure for the first time. To tackle the above challenge, we present LG-ODE, a latent ordinary differential equation generative model for modeling multi-agent dynamic system with known graph structure. It can simultaneously learn the embedding of high dimensional trajectories and infer continuous latent system dynamics. Our model employs a novel encoder parameterized by a graph neural network that can infer initial states in an unsupervised way from irregularly-sampled partial observations of structural objects and utilizes neuralODE to infer arbitrarily complex continuous-time latent dynamics. Experiments on motion capture, spring system, and charged particle datasets demonstrate the effectiveness of our approach.

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.