mlsanityData to 7 Oct 2026

Deep-OSG: Deep Learning of Operators in Semigroup

arXiv 2302.03358 ↗First posted 7 Feb 2023cs.LG cs.NA math.DScited by 9

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Abstract (arXiv)

This paper proposes a novel deep learning approach for learning operators in semigroup, with applications to modeling unknown autonomous dynamical systems using time series data collected at varied time lags. It is a sequel to the previous flow map learning (FML) works [T. Qin, K. Wu, and D. Xiu, J. Comput. Phys., 395:620--635, 2019], [K. Wu and D. Xiu, J. Comput. Phys., 408:109307, 2020], and [Z. Chen, V. Churchill, K. Wu, and D. Xiu, J. Comput. Phys., 449:110782, 2022], which focused on learning single evolution operator with a fixed time step. This paper aims to learn a family of evolution operators with variable time steps, which constitute a semigroup for an autonomous system. The semigroup property is very crucial and links the system's evolutionary behaviors across varying time scales, but it was not considered in the previous works. We propose for the first time a framework of embedding the semigroup property into the data-driven learning process, through a novel neural network architecture and new loss functions. The framework is very feasible, can be combined with any suitable neural networks, and is applicable to learning general autonomous ODEs and PDEs. We present the rigorous error estimates and variance analysis to understand the prediction accuracy and robustness of our approach, showing the remarkable advantages of semigroup awareness in our model. Moreover, our approach allows one to arbitrarily choose the time steps for prediction and ensures that the predicted results are well self-matched and consistent. Extensive numerical experiments demonstrate that embedding the semigroup property notably reduces the data dependency of deep learning models and greatly improves the accuracy, robustness, and stability for long-time prediction.

Key facts. Cited by 9 papers in the map (data through 7 Oct 2026). Builds on 27 of its 53 references that are in the map. First posted on arXiv 7 Feb 2023.

Builds on

27 of 53 references.

References · 27

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15 more
  1. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
  2. 2020DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithmcited by 45
  3. 2022Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Networkcited by 45
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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 2; 2025: 1; 2026: 5.

9 papers.

  1. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  2. 2023Improving physics-informed DeepONets with hard constraintscited by 9
  3. 2024A scalable generative model for dynamical system reconstruction from neuroimaging datacited by 9
  4. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  5. 2026Data-Driven Open-Loop Simulation for Digital-Twin Operator Decision Support in Wastewater Treatmentcited by 2
  6. 2026Semigroup Consistency as a Diagnostic for Learned Physics Simulatorscited by 2
  7. 2026NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equationscited by 0
  8. 2026Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training errorcited by 0
  9. 2026Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamicscited by 0