mlsanityData to 14 Oct 2026

Data Driven Governing Equations Approximation Using Deep Neural Networks

arXiv 1811.05537 ↗First posted 13 Nov 2018math.NA cs.LG cs.NAcited by 96

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Abstract (arXiv)

We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.

Key facts. Cited by 96 papers in the map (data through 14 Oct 2026). First posted on arXiv 13 Nov 2018.

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Citing papers per year: 2021: 27; 2022: 22; 2023: 14; 2024: 11; 2025: 10; 2026: 12.

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  1. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  2. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  3. 2021GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systemscited by 49
  4. 2021Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Mapscited by 41
  5. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  6. 2021Constructing Neural Network-Based Models for Simulating Dynamical Systemscited by 31
  7. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  8. 2021DAE-PINN: A Physics-Informed Neural Network Model for Simulating Differential-Algebraic Equations with Application to Power Networkscited by 28
  9. 2021Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Spacecited by 25
  10. 2023Learning Stochastic Dynamical System via Flow Map Operatorcited by 24
  11. 2025Reduced Order Modeling with Shallow Recurrent Decoder Networkscited by 24
  12. 2021Model-free inference of unseen attractors: Reconstructing phase space features from a single noisy trajectory using reservoir computingcited by 21
38 more
  1. 2024A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systemscited by 19
  2. 2022Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation datacited by 18
  3. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
  4. 2023Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarkscited by 18
  5. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  6. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  7. 2022DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-shot Transfer the Dynamic Response of Networked Systemscited by 14
  8. 2021A Deep Collocation Method for the Bending Analysis of Kirchhoff Platecited by 13
  9. 2021Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy datacited by 13
  10. 2022Online Weak-form Sparse Identification of Partial Differential Equationscited by 13
  11. 2023On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulatorscited by 13
  12. 2022PARC: Physics-Aware Recurrent Convolutional Neural Networks to Assimilate Meso-scale Reactive Mechanics of Energetic Materialscited by 11
  13. 2023Deep-OSG: Deep Learning of Operators in Semigroupcited by 11
  14. 2021Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama schemecited by 10
  15. 2022Physics-informed neural networks for solving parametric magnetostatic problemscited by 10
  16. 2021Model Order Reduction based on Runge-Kutta Neural Networkcited by 9
  17. 2021DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle methodcited by 9
  18. 2022Deep Learning of Chaotic Systems from Partially-Observed Datacited by 9
  19. 2023Finite Expression Methods for Discovering Physical Laws from Datacited by 9
  20. 2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operatorscited by 8
  21. 2022Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observationscited by 8
  22. 2025DUE: A Deep Learning Framework and Library for Modeling Unknown Equationscited by 8
  23. 2025FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learningcited by 8
  24. 2021GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equationscited by 7
  25. 2021Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systemscited by 7
  26. 2023Approaching epidemiological dynamics of COVID-19 with physics-informed neural networkscited by 7
  27. 2024Modeling Unknown Stochastic Dynamical System Subject to External Excitationcited by 7
  28. 2025Identifiability Challenges in Sparse Linear Ordinary Differential Equationscited by 7
  29. 2023InVAErt networks: a data-driven framework for model synthesis and identifiability analysiscited by 6
  30. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5
  31. 2024Sparse discovery of differential equations based on multi-fidelity Gaussian processcited by 5
  32. 2024Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systemscited by 5
  33. 2025Targeted Digital Twin via Flow Map Learning and Its Application to Fluid Dynamicscited by 5
  34. 2022Modeling unknown dynamical systems with hidden parameterscited by 4
  35. 2022Accelerating hydrodynamic simulations of urban drainage systems with physics-guided machine learningcited by 4
  36. 2022Learning nonparametric ordinary differential equations from noisy datacited by 4
  37. 2022SPADE4: Sparsity and Delay Embedding based Forecasting of Epidemicscited by 4
  38. 2023On the effectiveness of neural priors in modeling dynamical systemscited by 4

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