# Deep-OSG: Deep Learning of Operators in Semigroup

arXiv: https://arxiv.org/abs/2302.03358 · HTML: https://mlsanity.com/paper/2302.03358/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 53 references that are in the map.
- First posted on arXiv 7 Feb 2023.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Numerical Aspects for Approximating Governing Equations Using Data](https://mlsanity.com/paper/1809.09170/) (2018)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Learning reduced systems via deep neural networks with memory](https://mlsanity.com/paper/2003.09451/) (2020)
- [Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data](https://mlsanity.com/paper/2005.07916/) (2020)
- [Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data](https://mlsanity.com/paper/1905.10396/) (2019)
- [Discovery of Dynamics Using Linear Multistep Methods](https://mlsanity.com/paper/1912.12728/) (2019)
- [Data-driven discovery of free-form governing differential equations](https://mlsanity.com/paper/1910.05117/) (2019)
- [On generalized residue network for deep learning of unknown dynamical systems](https://mlsanity.com/paper/2002.02528/) (2020)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)

## Built on by (most cited first, 11 of 11)

- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)
- [A scalable generative model for dynamical system reconstruction from neuroimaging data](https://mlsanity.com/paper/2411.02949/) (2024)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Semigroup Consistency as a Diagnostic for Learned Physics Simulators](https://mlsanity.com/paper/2605.26324/) (2026)
- [Data-Driven Open-Loop Simulation for Digital-Twin Operator Decision Support in Wastewater Treatment](https://mlsanity.com/paper/2604.20935/) (2026)
- [NewPINNs: Physics-Informing Neural Networks Using Conventional Solvers for Partial Differential Equations](https://mlsanity.com/paper/2601.17207/) (2026)
- [Flow map learning in nonlinear vector autoregressive models: influence of the feature-library structure on the training error](https://mlsanity.com/paper/2605.31438/) (2026)
- [Physics-Constrained Neural Flow Maps for Long-Horizon Prediction of Spin Dynamics](https://mlsanity.com/paper/2608.22006/) (2026)
- [Physics is the Best Teacher: Consistency Learning for Time-Invariant Operators of Chaotic Dynamics](https://mlsanity.com/paper/2610.04108/) (2026)
- [Structure-preserving fidelity diagnostics for partial differential equations](https://mlsanity.com/paper/2610.17483/) (2026)

## 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.