Data to 7 Oct 2026

Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise

arXiv 1906.02355 ↗First posted 5 Jun 2019cs.LG cs.AI cs.CVcited by 98

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Abstract (arXiv)

Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous neural network framework called Neural Stochastic Differential Equation (Neural SDE) network, which naturally incorporates various commonly used regularization mechanisms based on random noise injection. Our framework can model various types of noise injection frequently used in discrete networks for regularization purpose, such as dropout and additive/multiplicative noise in each block. We provide theoretical analysis explaining the improved robustness of Neural SDE models against input perturbations/adversarial attacks. Furthermore, we demonstrate that the Neural SDE network can achieve better generalization than the Neural ODE and is more resistant to adversarial and non-adversarial input perturbations.

Key facts. Cited by 98 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Jun 2019.

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Citing papers per year: 2021: 19; 2022: 14; 2023: 15; 2024: 21; 2025: 21; 2026: 8.

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2021GRAND: Graph Neural Diffusioncited by 258
2021Neural SDEs as Infinite-Dimensional GANscited by 159
2021Beltrami Flow and Neural Diffusion on Graphscited by 84
2021Efficient and Accurate Gradients for Neural SDEscited by 68
2024KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physicscited by 58
2021Neural ODE Processescited by 57
2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 46
2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
2021Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equationscited by 38
2024Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Datacited by 35
2021Constructing Neural Network-Based Models for Simulating Dynamical Systemscited by 29
2021Neural Delay Differential Equationscited by 26

38 more

2021Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristicscited by 26
2025Comprehensive Review of Neural Differential Equations for Time Series Analysiscited by 20
2021Policy Analysis using Synthetic Controls in Continuous-Timecited by 18
2023Look Beneath the Surface: Exploiting Fundamental Symmetry for Sample-Efficient Offline RLcited by 18
2022TIDE: Time Derivative Diffusion for Deep Learning on Graphscited by 17
2022Learning Stochastic Dynamics with Statistics-Informed Neural Networkcited by 16
2023Latent SDEs on Homogeneous Spacescited by 16
2023From Coupled Oscillators to Graph Neural Networks: Reducing Over-smoothing via a Kuramoto Model-based Approachcited by 15
2021Neural network stochastic differential equation models with applications to financial data forecastingcited by 14
2023Improving Transferability for Cross-domain Trajectory Prediction via Neural Stochastic Differential Equationcited by 12
2022Fractional SDE-Net: Generation of Time Series Data with Long-term Memorycited by 9
2023Variational Inference for SDEs Driven by Fractional Noisecited by 9
2024From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systemscited by 9
2022TO-FLOW: Efficient Continuous Normalizing Flows with Temporal Optimization adjoint with Moving Speedcited by 8
2022Path Development Network with Finite-dimensional Lie Group Representationcited by 8
2022Learning Differential Operators for Interpretable Time Series Modelingcited by 7
2025In-Context Learning of Stochastic Differential Equations with Foundation Inference Modelscited by 7
2024Partially Stochastic Infinitely Deep Bayesian Neural Networkscited by 6
2022Continuous Deep Equilibrium Models: Training Neural ODEs faster by integrating them to Infinitycited by 5
2023Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-Differential Equationscited by 5
2025Robust Asymmetric Heterogeneous Federated Learning with Corrupted Clientscited by 5
2021Federated Learning as a Mean-Field Gamecited by 4
2022Neural Piecewise-Constant Delay Differential Equationscited by 4
2022Learning Modular Simulations for Homogeneous Systemscited by 4
2023Locally Regularized Neural Differential Equations: Some Black Boxes Were Meant to Remain Closed!cited by 4
2024A Cost-Efficient FPGA Implementation of Tiny Transformer Model using Neural ODEcited by 4
2024Universal randomised signatures for generative time series modellingcited by 4
2024Fully Bayesian Differential Gaussian Processes through Stochastic Differential Equationscited by 4
2024Deterministic versus stochastic dynamical classifiers: opposing random adversarial attacks with noisecited by 4
2025Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamicscited by 4
2026ODEWorld: A Continuous Predictive Architecture via Physical-Time Flowcited by 4
2021Robust and Scalable SDE Learning: A Functional Perspectivecited by 3
2022Mixed Effects Neural ODE: A Variational Approximation for Analyzing the Dynamics of Panel Datacited by 3
2022AdamNODEs: When Neural ODE Meets Adaptive Moment Estimationcited by 3
2023Learning to Decouple Complex Systemscited by 3
2023Faster Training of Neural ODEs Using Gauss-Legendre Quadraturecited by 3
2024Alternators For Sequence Modelingcited by 3
2025Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learningcited by 3

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