Generalization Bound for a General Class of Neural Ordinary Differential Equations

• URL: http://arxiv.org/abs/2508.18920v1
• Date: Tue, 26 Aug 2025 10:47:59 GMT
• Title: Generalization Bound for a General Class of Neural Ordinary Differential Equations
• Authors: Madhusudan Verma, Manoj Kumar,
• Abstract summary: We establish generalization bounds for both time-dependent and time-independent cases of neural ODEs.<n>This is the first derivation of generalization bounds for neural ODEs with general nonlinear dynamics.
• Score: 8.698347221292998
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural ordinary differential equations (neural ODEs) are a popular type of deep learning model that operate with continuous-depth architectures. To assess how well such models perform on unseen data, it is crucial to understand their generalization error bounds. Previous research primarily focused on the linear case for the dynamics function in neural ODEs - Marion, P. (2023), or provided bounds for Neural Controlled ODEs that depend on the sampling interval Bleistein et al. (2023). In this work, we analyze a broader class of neural ODEs where the dynamics function is a general nonlinear function, either time dependent or time independent, and is Lipschitz continuous with respect to the state variables. We showed that under this Lipschitz condition, the solutions to neural ODEs have solutions with bounded variations. Based on this observation, we establish generalization bounds for both time-dependent and time-independent cases and investigate how overparameterization and domain constraints influence these bounds. To our knowledge, this is the first derivation of generalization bounds for neural ODEs with general nonlinear dynamics.

Related papers

• Neural Time-Reversed Generalized Riccati Equation [60.92253836775246]
  Hamiltonian equations offer an interpretation of optimality through auxiliary variables known as costates. This paper introduces a novel neural-based approach to optimal control, with the aim of working forward-in-time.
  arXiv  Detail & Related papers  (2023-12-14T19:29:37Z)
• Individualized Dosing Dynamics via Neural Eigen Decomposition [51.62933814971523]
  We introduce the Neural Eigen Differential Equation algorithm (NESDE) NESDE provides individualized modeling, tunable generalization to new treatment policies, and fast, continuous, closed-form prediction. We demonstrate the robustness of NESDE in both synthetic and real medical problems, and use the learned dynamics to publish simulated medical gym environments.
  arXiv  Detail & Related papers  (2023-06-24T17:01:51Z)
• Generalization bounds for neural ordinary differential equations and deep residual networks [1.2328446298523066]
  We consider a family of parameterized neural ordinary differential equations (neural ODEs) with continuous-in-time parameters. By leveraging the analogy between neural ODEs and deep residual networks, our approach yields a generalization bound for a class of deep residual networks.
  arXiv  Detail & Related papers  (2023-05-11T08:29:34Z)
• Generalized Neural Closure Models with Interpretability [28.269731698116257]
  We develop a novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models.
  arXiv  Detail & Related papers  (2023-01-15T21:57:43Z)
• Neural Laplace: Learning diverse classes of differential equations in the Laplace domain [86.52703093858631]
  We propose a unified framework for learning diverse classes of differential equations (DEs) including all the aforementioned ones. Instead of modelling the dynamics in the time domain, we model it in the Laplace domain, where the history-dependencies and discontinuities in time can be represented as summations of complex exponentials. In the experiments, Neural Laplace shows superior performance in modelling and extrapolating the trajectories of diverse classes of DEs.
  arXiv  Detail & Related papers  (2022-06-10T02:14:59Z)
• Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems [1.001737665513683]
  We present a data-driven modeling method that accurately captures shocks and chaotic dynamics. We learn the right-hand-side (SRH) of an ODE by adding the outputs of two NN together where one learns a linear term and the other a nonlinear term. Specifically, we implement this by training a sparse linear convolutional NN to learn the linear term and a dense fully-connected nonlinear NN to learn the nonlinear term.
  arXiv  Detail & Related papers  (2022-03-29T16:10:34Z)
• Neural Stochastic Partial Differential Equations [1.2183405753834562]
  We introduce the Neural SPDE model providing an extension to two important classes of physics-inspired neural architectures. On the one hand, it extends all the popular neural -- ordinary, controlled, rough -- differential equation models in that it is capable of processing incoming information. On the other hand, it extends Neural Operators -- recent generalizations of neural networks modelling mappings between functional spaces -- in that it can be used to learn complex SPDE solution operators.
  arXiv  Detail & Related papers  (2021-10-19T20:35:37Z)
• 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)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.