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Universal Differential Equations for Scientific Machine Learning

arXiv 2001.04385 ↗First posted 13 Jan 2020cs.LG math.DS q-bio.QMcited by 345

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Abstract (arXiv)

In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." In this manuscript we introduce the SciML software ecosystem as a tool for mixing the information of physical laws and scientific models with data-driven machine learning approaches. We describe a mathematical object, which we denote universal differential equations (UDEs), as the unifying framework connecting the ecosystem. We show how a wide variety of applications, from automatically discovering biological mechanisms to solving high-dimensional Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled through the UDE formalism and its tooling. We demonstrate the generality of the software tooling to handle stochasticity, delays, and implicit constraints. This funnels the wide variety of SciML applications into a core set of training mechanisms which are highly optimized, stabilized for stiff equations, and compatible with distributed parallelism and GPU accelerators.

Key facts. Cited by 345 papers in the map (data through 7 Oct 2026). First posted on arXiv 13 Jan 2020.

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Citing papers per year: 2021: 53; 2022: 54; 2023: 43; 2024: 50; 2025: 69; 2026: 76.

345 papers.

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  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  3. 2021Neural SDEs as Infinite-Dimensional GANscited by 159
  4. 2022Monarch: Expressive Structured Matrices for Efficient and Accurate Trainingcited by 100
  5. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 90
  6. 2021Chaos as an interpretable benchmark for forecasting and data-driven modellingcited by 87
  7. 2021Stiff Neural Ordinary Differential Equationscited by 77
  8. 2022Rediscovering orbital mechanics with machine learningcited by 71
  9. 2022Human Trajectory Prediction via Neural Social Physicscited by 66
  10. 2021Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modelingcited by 59
  11. 2024KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physicscited by 58
  12. 2021Machine learning structure preserving brackets for forecasting irreversible processescited by 51
38 more
  1. 2021Neural Controlled Differential Equations for Online Prediction Taskscited by 46
  2. 2021Physics informed neural networks for continuum micromechanicscited by 46
  3. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  4. 2025Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraintscited by 46
  5. 2021Physics-Guided Deep Learning for Dynamical Systems: A Surveycited by 43
  6. 2021Neural Networks with Physics-Informed Architectures and Constraints for Dynamical Systems Modelingcited by 40
  7. 2021A Framework for Machine Learning of Model Error in Dynamical Systemscited by 37
  8. 2023Interpretable Machine Learning for Science with PySR and SymbolicRegression.jlcited by 36
  9. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  10. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  11. 2023Physics-Informed Machine Learning for Modeling and Control of Dynamical Systemscited by 28
  12. 2021Opening the Blackbox: Accelerating Neural Differential Equations by Regularizing Internal Solver Heuristicscited by 26
  13. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  14. 2023Reversible and irreversible bracket-based dynamics for deep graph neural networkscited by 25
  15. 2021Hybrid FEM-NN models: Combining artificial neural networks with the finite element methodcited by 24
  16. 2022Adaptive Self-supervision Algorithms for Physics-informed Neural Networkscited by 24
  17. 2022Interpretable Polynomial Neural Ordinary Differential Equationscited by 24
  18. 2022Differentiable, learnable, regionalized process-based models with physical outputs can approach state-of-the-art hydrologic prediction accuracycited by 21
  19. 2022Differentiable Programming for Earth System Modelingcited by 21
  20. 2022Guiding continuous operator learning through Physics-based boundary constraintscited by 21
  21. 2025Comprehensive Review of Neural Differential Equations for Time Series Analysiscited by 20
  22. 2021Stochastic Physics-Informed Neural Ordinary Differential Equationscited by 19
  23. 2021Physics-enhanced deep surrogates for partial differential equationscited by 18
  24. 2021Learning Stable Deep Dynamics Models for Partially Observed or Delayed Dynamical Systemscited by 17
  25. 2022Online Dynamics Learning for Predictive Control with an Application to Aerial Robotscited by 17
  26. 2022Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networkscited by 17
  27. 2024Mechanistic Neural Networks for Scientific Machine Learningcited by 16
  28. 2021Bridging observation, theory and numerical simulation of the ocean using Machine Learningcited by 15
  29. 2023Stabilized Neural Differential Equations for Learning Dynamics with Explicit Constraintscited by 15
  30. 2021Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolutioncited by 14
  31. 2021Finite Volume Neural Network: Modeling Subsurface Contaminant Transportcited by 14
  32. 2021Neural Network Verification in Controlcited by 14
  33. 2022Neural Ordinary Differential Equations for Nonlinear System Identificationcited by 14
  34. 2022Homotopy-based training of NeuralODEs for accurate dynamics discoverycited by 14
  35. 2024When are dynamical systems learned from time series data statistically accurate?cited by 14
  36. 2021Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy datacited by 13
  37. 2021Embedded training of neural-network sub-grid-scale turbulence modelscited by 13
  38. 2022Trust in AI: Interpretability is not necessary or sufficient, while black-box interaction is necessary and sufficientcited by 13

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