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Liquid Time-constant Networks

arXiv 2006.04439 ↗First posted 8 Jun 2020cs.LG cs.NE stat.MLcited by 161

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Abstract (arXiv)

We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems modulated via nonlinear interlinked gates. The resulting models represent dynamical systems with varying (i.e., liquid) time-constants coupled to their hidden state, with outputs being computed by numerical differential equation solvers. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations, and give rise to improved performance on time-series prediction tasks. To demonstrate these properties, we first take a theoretical approach to find bounds over their dynamics and compute their expressive power by the trajectory length measure in latent trajectory space. We then conduct a series of time-series prediction experiments to manifest the approximation capability of Liquid Time-Constant Networks (LTCs) compared to classical and modern RNNs. Code and data are available at https://github.com/raminmh/liquid_time_constant_networks

Key facts. Cited by 161 papers in the map (data through 14 Oct 2026). First posted on arXiv 8 Jun 2020.

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Citing papers per year: 2021: 13; 2022: 12; 2023: 21; 2024: 26; 2025: 33; 2026: 56.

161 papers.

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  1. 2021Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layerscited by 608
  2. 2023Resurrecting Recurrent Neural Networks for Long Sequencescited by 408
  3. 2022Liquid Structural State-Space Modelscited by 136
  4. 2024SiMBA: Simplified Mamba-Based Architecture for Vision and Multivariate Time seriescited by 82
  5. 2021Closed-form Continuous-time Neural Modelscited by 65
  6. 2024Mamba-360: Survey of State Space Models as Transformer Alternative for Long Sequence Modelling: Methods, Applications, and Challengescited by 64
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  8. 2021Accurate online training of dynamical spiking neural networks through Forward Propagation Through Timecited by 34
  9. 2021Latent Imagination Facilitates Zero-Shot Transfer in Autonomous Racingcited by 26
  10. 2024Trajectory Flow Matching with Applications to Clinical Time Series Modelingcited by 24
  11. 2021Causal Navigation by Continuous-time Neural Networkscited by 23
  12. 2025Continuous Thought Machinescited by 22
38 more
  1. 2026Large Language Model Reasoning Failurescited by 21
  2. 2021Adversarial Training is Not Ready for Robot Learningcited by 20
  3. 2024S7: Selective and Simplified State Space Layers for Sequence Modelingcited by 17
  4. 2024Rough Transformers: Lightweight and Continuous Time Series Modelling through Signature Patchingcited by 16
  5. 2025A Survey of Robotic Navigation and Manipulation with Physics Simulators in the Era of Embodied AIcited by 16
  6. 2021On-Off Center-Surround Receptive Fields for Accurate and Robust Image Classificationcited by 15
  7. 2024Neural Eulerian Scene Flow Fieldscited by 15
  8. 2022Homotopy-based training of NeuralODEs for accurate dynamics discoverycited by 14
  9. 2022Evolution of Neural Tangent Kernels under Benign and Adversarial Trainingcited by 14
  10. 2024Gaussian Splatting to Real World Flight Navigation Transfer with Liquid Networkscited by 14
  11. 2021Sparse Flows: Pruning Continuous-depth Modelscited by 13
  12. 2021GoTube: Scalable Stochastic Verification of Continuous-Depth Modelscited by 12
  13. 2021BarrierNet: A Safety-Guaranteed Layer for Neural Networkscited by 12
  14. 2022Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solverscited by 12
  15. 2024The Belief State Transformercited by 12
  16. 2022On the Forward Invariance of Neural ODEscited by 11
  17. 2022Differentiable Control Barrier Functions for Vision-based End-to-End Autonomous Drivingcited by 10
  18. 2025Trustless Autonomy: Understanding Motivations, Benefits, and Governance Dilemmas in Self-Sovereign Decentralized AI Agentscited by 10
  19. 2025Parallelization of Non-linear State-Space Models: Scaling Up Liquid-Resistance Liquid-Capacitance Networks for Efficient Sequence Modelingcited by 10
  20. 2023ODE-based Recurrent Model-free Reinforcement Learning for POMDPscited by 7
  21. 2025Before It's Too Late: A State Space Model for the Early Prediction of Misinformation and Disinformation Engagementcited by 6
  22. 2025Spore in the Wild: A Case Study of Spore.fun as an Open-Environment Evolution Experiment with Sovereign AI Agents on TEE-Secured Blockchainscited by 6
  23. 2026Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material modelscited by 6
  24. 2022Are All Vision Models Created Equal? A Study of the Open-Loop to Closed-Loop Causality Gapcited by 5
  25. 2023Stable Attractors for Neural networks classification via Ordinary Differential Equations (SA-nODE)cited by 5
  26. 2023Learning with Chemical versus Electrical Synapses -- Does it Make a Difference?cited by 5
  27. 2024Physical Reservoir Computing Enabled by Solitary Waves and Biologically-Inspired Nonlinear Transformation of Input Datacited by 5
  28. 2024Heracles: A Hybrid SSM-Transformer Model for High-Resolution Image and Time-Series Analysiscited by 5
  29. 2026Positive Alignment: Artificial Intelligence for Human Flourishingcited by 5
  30. 2022Entangled Residual Mappingscited by 4
  31. 2024CPS-LLM: Large Language Model based Safe Usage Plan Generator for Human-in-the-Loop Human-in-the-Plant Cyber-Physical Systemcited by 4
  32. 2024Deterministic versus stochastic dynamical classifiers: opposing random adversarial attacks with noisecited by 4
  33. 2024Benchmarking Deep Reinforcement Learning for Navigation in Denied Sensor Environmentscited by 4
  34. 2024Recovering implicit physics model under real-world constraintscited by 4
  35. 2025A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strainscited by 4
  36. 2023Branched Latent Neural Mapscited by 3
  37. 2024What comes after transformers? -- A selective survey connecting ideas in deep learningcited by 3
  38. 2024Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiologycited by 3

50 most cited of 161 All 161 (JSON)