Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations

• URL: http://arxiv.org/abs/2507.19036v1
• Date: Fri, 25 Jul 2025 07:44:34 GMT
• Title: Neural Ordinary Differential Equations for Learning and Extrapolating System Dynamics Across Bifurcations
• Authors: Eva van Tegelen, George van Voorn, Ioannis Athanasiadis, Peter van Heijster,
• Abstract summary: We develop a continuous, data-driven framework for learning system dynamics.<n>We show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from timeseries data.<n>We also assess the method's performance under limited and noisy data conditions.
• Score: 1.4277745078693944
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Forecasting system behaviour near and across bifurcations is crucial for identifying potential shifts in dynamical systems. While machine learning has recently been used to learn critical transitions and bifurcation structures from data, most studies remain limited as they exclusively focus on discrete-time methods and local bifurcations. To address these limitations, we use Neural Ordinary Differential Equations which provide a continuous, data-driven framework for learning system dynamics. We apply our approach to a predator-prey system that features both local and global bifurcations, presenting a challenging test case. Our results show that Neural Ordinary Differential Equations can recover underlying bifurcation structures directly from timeseries data by learning parameter-dependent vector fields. Notably, we demonstrate that Neural Ordinary Differential Equations can forecast bifurcations even beyond the parameter regions represented in the training data. We also assess the method's performance under limited and noisy data conditions, finding that model accuracy depends more on the quality of information that can be inferred from the training data, than on the amount of data available.

Related papers

• Disordered Dynamics in High Dimensions: Connections to Random Matrices and Machine Learning [52.26396748560348]
  We provide an overview of high dimensional dynamical systems driven by random matrices.<n>We focus on applications to simple models of learning and generalization in machine learning theory.
  arXiv  Detail & Related papers  (2026-01-03T00:12:32Z)
• High-order expansion of Neural Ordinary Differential Equations flows [4.4569182855550755]
  We introduce Event Transitions, a framework based on high-order differentials that provides a rigorous mathematical description of neural ODE dynamics on event gradient.<n>Our findings contribute to a deeper theoretical foundation for event-triggered neural differential equations and provide a mathematical construct for explaining complex system dynamics.
  arXiv  Detail & Related papers  (2025-04-02T08:57:34Z)
• Capturing the Temporal Dependence of Training Data Influence [100.91355498124527]
  We formalize the concept of trajectory-specific leave-one-out influence, which quantifies the impact of removing a data point during training.<n>We propose data value embedding, a novel technique enabling efficient approximation of trajectory-specific LOO.<n>As data value embedding captures training data ordering, it offers valuable insights into model training dynamics.
  arXiv  Detail & Related papers  (2024-12-12T18:28:55Z)
• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Let's do the time-warp-attend: Learning topological invariants of dynamical systems [3.9735602856280132]
  We propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a range of applications. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.
  arXiv  Detail & Related papers  (2023-12-14T18:57:16Z)
• Towards stable real-world equation discovery with assessing differentiating quality influence [52.2980614912553]
  We propose alternatives to the commonly used finite differences-based method. We evaluate these methods in terms of applicability to problems, similar to the real ones, and their ability to ensure the convergence of equation discovery algorithms.
  arXiv  Detail & Related papers  (2023-11-09T23:32:06Z)
• Learning invariant representations of time-homogeneous stochastic dynamical systems [27.127773672738535]
  We study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system. We show that the search for a good representation can be cast as an optimization problem over neural networks.
  arXiv  Detail & Related papers  (2023-07-19T11:32:24Z)
• Learning Latent Dynamics via Invariant Decomposition and (Spatio-)Temporal Transformers [0.6767885381740952]
  We propose a method for learning dynamical systems from high-dimensional empirical data. We focus on the setting in which data are available from multiple different instances of a system. We study behaviour through simple theoretical analyses and extensive experiments on synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2023-06-21T07:52:07Z)
• Stretched and measured neural predictions of complex network dynamics [2.1024950052120417]
  Data-driven approximations of differential equations present a promising alternative to traditional methods for uncovering a model of dynamical systems. A recently employed machine learning tool for studying dynamics is neural networks, which can be used for data-driven solution finding or discovery of differential equations. We show that extending the model's generalizability beyond traditional statistical learning theory limits is feasible.
  arXiv  Detail & Related papers  (2023-01-12T09:44:59Z)
• A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization [1.2882319878552302]
  This paper develops a new iterative learning algorithm for complex turbulent systems with partial observations. It alternates between identifying model structures, recovering unobserved variables, and estimating parameters. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable parameterizations for many complex nonlinear systems.
  arXiv  Detail & Related papers  (2022-08-19T00:35:03Z)
• Semi-supervised Learning of Partial Differential Operators and Dynamical Flows [68.77595310155365]
  We present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is able to interpolate and the solutions to any intermediate time.
  arXiv  Detail & Related papers  (2022-07-28T19:59:14Z)
• Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations [68.62843292346813]
  We propose a structured latent ODE model that captures system input variations within its latent representation. Building on a static variable specification, our model learns factors of variation for each input to the system, thus separating the effects of the system inputs in the latent space.
  arXiv  Detail & Related papers  (2022-02-25T20:00:56Z)
• Data-driven emergence of convolutional structure in neural networks [83.4920717252233]
  We show how fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs. By carefully designing data models, we show that the emergence of this pattern is triggered by the non-Gaussian, higher-order local structure of the inputs.
  arXiv  Detail & Related papers  (2022-02-01T17:11:13Z)
• Learning Neural Causal Models with Active Interventions [83.44636110899742]
  We introduce an active intervention-targeting mechanism which enables a quick identification of the underlying causal structure of the data-generating process. Our method significantly reduces the required number of interactions compared with random intervention targeting. We demonstrate superior performance on multiple benchmarks from simulated to real-world data.
  arXiv  Detail & Related papers  (2021-09-06T13:10:37Z)
• Supervised DKRC with Images for Offline System Identification [77.34726150561087]
  Modern dynamical systems are becoming increasingly non-linear and complex. There is a need for a framework to model these systems in a compact and comprehensive representation for prediction and control. Our approach learns these basis functions using a supervised learning approach.
  arXiv  Detail & Related papers  (2021-09-06T04:39:06Z)
• A Framework for Machine Learning of Model Error in Dynamical Systems [7.384376731453594]
  We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from data. We cast the problem in both continuous- and discrete-time, for problems in which the model error is memoryless and in which it has significant memory. We find that hybrid methods substantially outperform solely data-driven approaches in terms of data hunger, demands for model complexity, and overall predictive performance.
  arXiv  Detail & Related papers  (2021-07-14T12:47:48Z)
• Consistency of mechanistic causal discovery in continuous-time using Neural ODEs [85.7910042199734]
  We consider causal discovery in continuous-time for the study of dynamical systems. We propose a causal discovery algorithm based on penalized Neural ODEs.
  arXiv  Detail & Related papers  (2021-05-06T08:48:02Z)
• Knowledge-Based Learning of Nonlinear Dynamics and Chaos [3.673994921516517]
  We present a universal learning framework for extracting predictive models from nonlinear systems based on observations. Our framework can readily incorporate first principle knowledge because it naturally models nonlinear systems as continuous-time systems.
  arXiv  Detail & Related papers  (2020-10-07T13:50:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.