Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis
Not on the map. Why?
Abstract (arXiv)
In this paper, the problem of data-driven discovery of nonlinear ordinary differential equations (ODEs) is recast, and a new interpretable machine learning (ML) method is proposed. The proposed method aims to learn the unknown vector field of nonlinear dynamics without prior knowledge of the system's physics from only one single state trajectory's data. The proposed method has two fundamental differences with existing methods: 1) the formulation presented in this method is derived based on Functional Analysis and Operator Theory, and 2) the cost function is constructed in the function space as a distance between two functions as an integral, instead of the discrete-sum of errors used in existing ML approaches. An incremental learning algorithm is proposed to learn the unknown vector field to handle new training samples in an online manner. The proposed method can discover the unknown vector field from both forced and unforced autonomous and non-autonomous (or time-varying) dynamical systems. The proposed method is able to simultaneously discover unknown external forces as a function of time and unknown underlying dynamics. Finally, numerical examples are given to demonstrate the advantages of the proposed method.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 49 of its 127 references that are in the map. First posted on arXiv 3 Sep 2026.
Builds on
49 of 127 references.
References · 49
- 2017Attention Is All You Needcited by 60,361
- 2018Neural Ordinary Differential Equationscited by 3,779
- 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
- 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
- 2018Continual Lifelong Learning with Neural Networks: A Reviewcited by 1,330
- 2023A Comprehensive Survey of Continual Learning: Theory, Method and Applicationcited by 872
- 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
- 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
- 2020Class-incremental learning: survey and performance evaluation on image classificationcited by 449
- 2021A Survey of Transformerscited by 401
- 2019Augmented Neural ODEscited by 397
- 2021Interpretable Machine Learning: Fundamental Principles and 10 Grand Challengescited by 367
37 more
- 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
- 2019Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systemscited by 205
- 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 137
- 2022Interpretable Scientific Discovery with Symbolic Regression: A Reviewcited by 105
- 2021Neural Flows: Efficient Alternative to Neural ODEscited by 101
- 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
- 2023A Mathematical Guide to Operator Learningcited by 76
- 2022Symbolic Regression is NP-hardcited by 71
- 2020On Second Order Behaviour in Augmented Neural ODEscited by 68
- 2020Deep Polynomial Neural Networkscited by 63
- 2020STEER: Simple Temporal Regularization For Neural ODEscited by 58
- 2023Generalized Teacher Forcing for Learning Chaotic Dynamicscited by 58
- 2016Generalizing Koopman Theory to allow for inputs and controlcited by 53
- 2021Machine learning structure preserving brackets for forecasting irreversible processescited by 52
- 2020Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Datacited by 37
- 2023Deep networks for system identification: a Surveycited by 37
- 2021Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approachcited by 36
- 2022Residual Dynamic Mode Decomposition: Robust and verified Koopmanismcited by 36
- 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
- 2021Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modelingcited by 27
- 2023Predicting Ordinary Differential Equations with Transformerscited by 26
- 2022Neural Laplace: Learning diverse classes of differential equations in the Laplace domaincited by 24
- 2022Interpretable Polynomial Neural Ordinary Differential Equationscited by 24
- 2017Online dynamic mode decomposition for time-varying systemscited by 23
- 2017Koopman-based lifting techniques for nonlinear systems identificationcited by 18
- 2020Revealing hidden dynamics from time-series data by ODENetcited by 17
- 2021A toolkit for data-driven discovery of governing equations in high-noise regimescited by 15
- 2022Homotopy-based training of NeuralODEs for accurate dynamics discoverycited by 14
- 2021Cardinality Minimization, Constraints, and Regularization: A Surveycited by 12
- 2022Koopman Form of Nonlinear Systems with Inputscited by 11
- 2022Deep Koopman Learning of Nonlinear Time-Varying Systemscited by 11
- 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
- 2022Sparse dynamical system identification with simultaneous structural parameters and initial condition estimationcited by 7
- 2023Bayesian polynomial neural networks and polynomial neural ordinary differential equationscited by 6
- 2023ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Predictioncited by 5
- 2018A Distributed Algorithm for Solving Linear Algebraic Equations Over Random Networkscited by 2
- 2026Symbolic Regression via Neural Networkscited by 2
Built on by
Not cited yet.