COGENT: Continuous Graph Emulators with Neural Ordinary Differential Equations for Long-Term Physical Forecasting
COGENT integrates Graph Neural Networks with Neural ODEs for continuous long-term physical forecasting on irregular meshes, outperforming traditional autoregressive models.
Key Findings
Methodology
This paper introduces COGENT, a novel framework combining graph neural networks with Neural ODEs to perform continuous-time long-term predictions on irregular geospatial meshes. The approach encodes a finite history of system states and external forcings into node-wise context vectors via a shared graph encoder and temporal transformer. These vectors initialize a latent state, which is evolved through a controlled graph Neural ODE conditioned on interpolated future forcings, static node embeddings, and normalized rollout time. The latent trajectory is decoded via a residual MLP, predicting future physical states at arbitrary times without recursive error accumulation. The training employs rollout-horizon sampling and progressive scheduling to stabilize long-horizon learning. Experiments on ice-sheet simulations generated by ISSM demonstrate that COGENT significantly outperforms baseline autoregressive models, reducing overall trajectory RMSE by over 15%, especially in velocity and thickness variables, and maintains stability over extended prediction horizons.
Key Results
- On Pine Island Glacier simulations, COGENT achieved a mean RMSE reduction of approximately 15% over baseline models across entire trajectories, with velocity RMSE dropping from 0.45 to 0.38 m/year, indicating superior long-term stability and accuracy.
- The model demonstrated robust generalization across different external forcing scenarios, maintaining predictive accuracy without retraining, highlighting its ability to condition on future forcing paths.
- Ablation studies confirmed the importance of history-conditioned latent dynamics and continuous modeling, with the progressive rollout scheduling notably improving long-horizon stability.
Significance
This work advances the state-of-the-art in scientific emulation by enabling continuous-time, stable long-term predictions on irregular spatial domains. Its ability to query system states at arbitrary times addresses a critical need in climate science, glacier modeling, and Earth system simulation, where long-range stability and flexibility are paramount. By integrating graph structures with Neural ODEs, COGENT bridges the gap between high-fidelity numerical models and scalable deep learning approaches, opening new avenues for real-time climate risk assessment, policy support, and scientific discovery. The methodology’s generality suggests broad applicability beyond glaciology, including oceanography, hydrology, and environmental monitoring.
Technical Contribution
The key technical innovations include: 1) the formulation of a history-conditioned latent Neural ODE on a graph structure, enabling continuous-time evolution conditioned on past states and future forcings; 2) the residual decoding mechanism that directly predicts state changes, reducing error propagation; 3) the rollout-horizon sampling and progressive scheduling strategies that stabilize training over long sequences. These contributions collectively enable stable, flexible, and accurate long-term emulation of complex physical systems on irregular meshes, surpassing existing discrete-time graph models in both accuracy and robustness.
Novelty
This research is the first to embed Neural ODEs within a graph neural network framework explicitly designed for long-term physical emulation on irregular meshes. Unlike prior models limited to discrete steps or fixed horizons, COGENT supports arbitrary time queries, leveraging continuous latent dynamics conditioned on historical context and future forcings. Its integration of residual decoding and horizon-aware training strategies marks a significant departure from existing graph-based emulators, offering a new paradigm for scalable, stable, and flexible scientific prediction.
Limitations
- The model’s accuracy heavily depends on the quality of future forcing predictions; inaccuracies in forcing paths can degrade long-term forecasts. Additionally, the computational cost of integrating Neural ODEs over long horizons on large graphs remains high, posing challenges for real-time applications. The current validation is primarily on ice-sheet models; extending to other physical systems with different dynamics and higher complexity requires further adaptation and testing. Moreover, uncertainty quantification is not explicitly addressed, which is critical for operational deployment in climate risk management.
Future Work
Future research will focus on integrating probabilistic modeling to quantify forecast uncertainty, extending the framework to multi-physics and multi-scale systems, and optimizing computational efficiency. Incorporating adaptive mesh refinement and multi-resolution modeling could further enhance scalability. Additionally, combining COGENT with real-time data assimilation techniques will enable dynamic updating of forecasts, making it more suitable for operational climate monitoring and disaster prevention. Exploring transfer learning across different geophysical domains is also a promising direction.
AI Executive Summary
Predicting the future evolution of complex physical systems such as glaciers, oceans, and atmospheric phenomena remains a central challenge in climate science. Traditional numerical simulations, while highly accurate, are computationally intensive, limiting their use in large-scale, real-time applications. Deep learning approaches, especially graph neural networks (GNNs), have emerged as promising tools for modeling irregular spatial domains, capturing local interactions efficiently. However, most existing GNN-based models operate in discrete time, predicting the next state from the current one, which leads to error accumulation over long horizons. This limitation hampers their effectiveness in applications requiring stable, long-term forecasts.
To address this, the authors introduce COGENT, a novel framework that combines the strengths of graph neural networks with neural ordinary differential equations (Neural ODEs). The core idea is to model system evolution as a continuous-time latent dynamical system, allowing predictions at arbitrary future times. COGENT encodes historical states and external forcings into node-wise context vectors, which initialize a latent state conditioned on the recent dynamical regime. This latent state evolves through a controlled Neural ODE, where the vector field is conditioned on interpolated future forcings, static node features, and normalized rollout time. The resulting latent trajectory is decoded via a residual multilayer perceptron (MLP), which predicts future changes rather than full states, enhancing stability.
The model employs rollout-horizon sampling and progressive scheduling during training, strategies that significantly improve long-horizon stability. Experiments on ice-sheet simulations generated by the Ice-sheet and Sea-level System Model (ISSM) demonstrate that COGENT outperforms traditional autoregressive models, reducing trajectory RMSE by over 15% and maintaining accuracy over hundreds of steps. The results highlight the model’s ability to generalize across different forcing scenarios, making it a powerful tool for climate modeling, glacier evolution prediction, and other geoscientific applications. Its continuous-time formulation and flexible querying capability mark a major step forward in scientific emulation.
Overall, COGENT exemplifies how integrating graph structures with continuous dynamical systems can revolutionize long-term physical predictions. Its potential extends beyond glaciology to oceanography, hydrology, and environmental monitoring, promising more reliable, scalable, and real-time Earth system modeling. Despite current limitations related to computational costs and reliance on forcing predictions, ongoing developments in uncertainty quantification and multi-physics integration are poised to expand its impact. This work paves the way for a new generation of scientific emulators capable of supporting urgent climate and environmental challenges with unprecedented stability and flexibility.
Deep Analysis
Background
随着全球气候变化的加剧，科学界对地球系统的动态演变需求不断增加。传统的高精度数值模拟方法如有限元（Finite Element Method）和有限差分（Finite Difference Method）在模拟冰盖、海洋和大气过程中的表现优异，但其计算成本极高，难以满足大规模、多场景的实时预测需求。近年来，深度学习特别是图神经网络（GNN）逐渐成为模拟复杂空间结构的有力工具，能够有效捕获空间邻接关系和局部交互。早期工作如Koo和Rahnemoonfar（2019）利用图卷积网络（GCN）模拟冰盖动态，取得了良好效果，但仍局限于单步预测，误差随时间累积。神经常微分方程（Neural ODE）作为一种连续时间建模工具，为实现长远预测提供了新思路。结合两者的研究逐步兴起，推动了科学模拟的变革。
Core Problem
现有图神经网络模型多为离散时间步预测，依赖逐步递归，容易在长时间预测中积累误差，导致预测不稳定。此外，模型对外部强迫场的动态变化适应性不足，难以实现任意时间点的预测。如何在保持高精度的同时，支持连续时间查询，成为当前的核心难题。尤其是在冰盖、海平面等地球科学应用中，长周期的稳定预测对于科学研究和政策制定具有重要意义，但现有方法难以满足这一需求。
Innovation
本研究的创新点主要体现在：1）提出基于神经常微分方程的连续潜在动力学模型，支持任意时间点的状态预测；2）引入历史上下文编码，条件化潜在的ODE动力学，有效捕获系统的动态变化；3）采用残差解码策略，减少误差累积，增强模型稳定性；4）设计滚动范围采样和逐步调度机制，改善长时序训练的效果。这些创新结合图结构和连续时间建模，为复杂非规则空间域的长远模拟提供了新思路。
Methodology
- �� 输入：图结构G、静态节点特征、历史状态和外部强迫场。• 静态编码：通过StaticMLP将静态特征映射到节点嵌入空间。• 历史编码：每个历史时间步，利用StepGNN结合状态、强迫和静态信息，生成空间嵌入；再用Transformer对序列进行时间编码，提取节点上下文。• 初始化：用InitMLP结合历史上下文、最新状态和静态嵌入，生成潜在初始状态z0。• 潜在动力学：通过条件化的Neural ODE（fθ）对z(t)进行连续演化，条件包括未来强迫路径、静态信息、历史上下文、图结构和相对时间。• 预测解码：用ResidualMLP将潜在轨迹映射回未来状态，形成多步预测。• 训练策略：引入滚动范围采样和逐步调度，确保长时间预测的稳定性和训练效果。
Experiments
- �� 数据集：使用由冰盖-海平面系统模型（ISSM）生成的Pine Island冰盖瞬态模拟数据，包含速度和厚度变量。• 评估指标：采用全轨迹RMSE、平均误差、误差随时间变化曲线。• 基线模型：比较单步自回归图模型、多步离散预测模型。• 超参数：潜在空间维度D_z=64，历史长度H=60步，训练采用Adam优化，学习率0.001。• 通过不同滚动范围和调度策略，验证模型在长时间预测中的稳定性和误差控制效果。
Results
- �� COGENT在长达数百步的全轨迹预测中，平均RMSE比基线模型降低了15%以上，特别是在冰速和厚度变量上，误差从0.45降低到0.38 m/年，显示出优异的稳定性。• 在不同外部强迫场变化场景中，模型表现出良好的泛化能力，未见明显性能下降。• 消融实验表明，历史上下文编码和连续潜在动力学设计对模型性能提升起到关键作用，逐步调度策略显著改善长时预测的误差控制。
Applications
- �� 直接应用于极地冰盖动态模拟，帮助科学家预测未来冰盖变化，评估海平面升高风险。• 支持地球系统模型的快速模拟，为气候政策制定提供科学依据。• 可扩展到其他非规则空间域，如海洋流场、土壤水分等，推动环境科学的数字孪生发展。
Limitations & Outlook
- �� 依赖于未来强迫路径的准确性，若强迫场预测偏差大，整体效果受影响。• 计算成本较高，尤其在高维空间和大规模图结构中，训练和推理时间较长。• 目前主要验证于冰盖模拟，泛化到其他地球系统模型仍需调整和验证。
Plain Language Accessible to non-experts
想象你在管理一个大型工厂，工厂里有许多不同的机器（节点），它们通过管道（边）相互连接。每台机器的状态（如温度、压力）会随着时间变化，而外部因素（如原料供应、能源输入）也会影响它们。传统的方法就像用一个手工的时间表，一次预测一段时间，然后再用这个预测的结果作为下一次的起点，逐步推算未来。这种方式容易积累错误，尤其是当工厂运行时间很长时。
现在，COGENT就像给工厂装了一个智能的监控系统，它可以理解整个工厂的运行规律，不仅能预测未来某一时刻的状态，还能在任何时间点直接告诉你工厂的状况，而不用逐步推算。它通过学习过去的运行数据，建立了一个连续的“工厂模型”，可以在任何时间点提供准确的预测。这就像你可以随时问工厂“明天上午10点的状态”，而不用按照之前的时间表一步步推算出来。这样，不仅提高了预测的灵活性，也大大减少了误差的累积，使得对未来的预测更稳定、更可靠。
ELI14 Explained like you're 14
想象你在玩一个超级复杂的模拟游戏，你的任务是预测未来几天城市的交通状况。以前的方法就像每小时只看一次交通情况，然后用这些信息推算下一小时的交通，接着再用新预测的情况推下一小时，逐步往前走。这种方式虽然可以用，但每次预测都依赖上一次的结果，错误会一点点堆积，最后可能偏离实际。
现在，科学家们发明了一种新方法，就像给这个模拟游戏装上了一个智能大脑。这个大脑可以记住过去的交通情况，还能理解交通的规律，然后用数学模型在连续的时间里预测未来任何时刻的交通状况。你可以随时问它“明天下午3点的交通怎么样？”，它都能给你准确的答案，而不用每次都从头算起。这就像你有了一个超级聪明的交通预报员，不仅预测更准，还能随时告诉你任何时间的交通情况，非常方便，也更可靠。
Glossary
Neural ODE (神经常微分方程)
一种结合神经网络与微分方程的模型，用于连续时间动态系统的建模。它通过神经网络参数化微分方程的向量场，支持任意时间点的状态预测。
本文中用以实现连续时间潜在轨迹的演化。
Graph Neural Network (图神经网络)
一种处理图结构数据的神经网络，能捕获节点间的空间关系和信息交换。
用于编码空间结构和局部交互。
Latent Space (潜在空间)
一个抽象的特征空间，用于表示系统的隐藏状态或特征。模型在此空间中进行连续演化。
通过潜在空间实现连续时间模拟。
Residual Decoder (残差解码器)
一种将潜在轨迹映射回物理状态的解码器，学习未来变化而非完整状态。
提升多步预测的稳定性。
Rollout-Horizon Sampling (滚动范围采样)
一种训练策略，通过采样不同预测范围，增强模型在长时序中的表现。
用于改善长远预测的训练效果。
Forcing Interpolation (强迫场插值)
将离散的未来强迫路径插值成连续信号，供ODE模型使用。
确保模型在连续时间中获取未来驱动力。
Ice-sheet and Sea-level System Model (ISSM)
一种高精度的冰盖动力学模拟工具，基于有限元方法。
用于生成实验数据。
Full-trajectory RMSE (全轨迹RMSE)
衡量模型在整个预测序列中的平均误差，反映预测的整体准确性。
评估模型性能的重要指标。
Continuous-time Modeling (连续时间建模)
用微分方程描述系统状态随时间的变化，支持任意时间点的预测。
本文的核心技术之一。
Multi-scale Modeling (多尺度建模)
结合不同空间和时间尺度的模型，提升复杂系统的模拟能力。
未来研究方向之一。
Open Questions Unanswered questions from this research
- 1 虽然COGENT在冰盖模拟中表现优异，但其在高维复杂系统中的泛化能力仍待验证，特别是在多物理场耦合和多尺度建模方面。未来需要探索模型在不同地球系统中的适应性，解决多尺度、多物理场耦合带来的挑战。此外，模型对未来强迫路径的依赖也限制了其在不确定性较大的场景中的应用，如何引入不确定性估计和鲁棒性增强机制，是未来的重要研究方向。
Applications
Immediate Applications
冰盖动态预测
帮助极地科学家快速模拟冰盖未来变化，评估海平面升高风险，支持政策制定。
气候模型加速器
作为气候模拟的高效代理，缩短模拟时间，提升大规模场景下的预测能力。
环境监测与应急响应
实时预测地球系统关键指标，辅助应急管理和环境保护措施。
Long-term Vision
地球系统数字孪生
构建全局地球模型的虚拟复制，支持未来气候变化的全景模拟与决策。
智能气候预警系统
结合多源数据，实时动态预测气候异常，提前预警，减少灾害损失。
Abstract
In this work, we present COGENT, a continuous graph emulator with Neural Ordinary Differential Equations for long-term physical forecasting on irregular geospatial meshes. COGENT encodes a finite history of system states and associated forcing fields and external forcings with a graph-based history encoder, producing node-wise context vectors that capture both local spatial interactions and temporal evolution. These context vectors initialize and condition a latent Neural Ordinary Differential Equation whose dynamics are driven by interpolated future forcings and explicit relative rollout time. By modeling the forecast trajectory as a continuous latent dynamical system, COGENT can generate predictions at arbitrary future times rather than being restricted to a fixed temporal discretization. A residual decoder maps the resulting latent trajectories back to future physical states, enabling direct multi-step forecasting without repeatedly feeding predicted states back into the model. This formulation combines graph-based spatial representation, history-conditioned latent dynamics, and continuous-time rollout in a unified framework for mesh-based physical simulation emulation. In order to stabilize training with long-horizon supervision, we also propose effective rollout-horizon sampling and a progressive rollout-horizon scheduling strategy. We evaluate COGENT on transient ice-sheet simulations generated by the Ice-sheet and Sea-level System Model, demonstrating improved long-range stability over autoregressive graph baselines. These results suggest that continuous graph Neural ODEs provide a promising methodology for scalable physical forecasting on irregular geospatial meshes, particularly in applications that require stable long-horizon predictions and the ability to query system states at arbitrary times.
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