mlsanityData to 7 Oct 2026

DeepMoD: Deep learning for Model Discovery in noisy data

arXiv 1904.09406 ↗First posted 20 Apr 2019physics.comp-ph q-bio.QM stat.MLcited by 70

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Abstract (arXiv)

We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, but it also performs the sparse regression. This construction makes it extremely robust to noise, applicable to small data sets, and, contrary to other deep learning methods, does not require a training set. We benchmark our approach on several physical problems such as the Burgers', Korteweg-de Vries and Keller-Segel equations, and find that it requires as few as O(102) samples and works at noise levels up to 75%. Motivated by these results, we apply DeepMoD directly on noisy experimental time-series data from a gel electrophoresis experiment and find that it discovers the advection-diffusion equation describing this system.

Key facts. Cited by 70 papers in the map (data through 7 Oct 2026). First posted on arXiv 20 Apr 2019.

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Citing papers per year: 2021: 15; 2022: 11; 2023: 10; 2024: 7; 2025: 11; 2026: 16.

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  1. 2022Rediscovering orbital mechanics with machine learningcited by 71
  2. 2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
  3. 2023Interpretable Machine Learning for Science with PySR and SymbolicRegression.jlcited by 36
  4. 2021PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learningcited by 24
  5. 2024SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learningcited by 24
  6. 2021Robust discovery of partial differential equations in complex situationscited by 21
  7. 2022PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Datacited by 20
  8. 2021Physics-informed Spline Learning for Nonlinear Dynamics Discoverycited by 18
  9. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  10. 2023Weak-PDE-LEARN: A Weak Form Based Approach to Discovering PDEs From Noisy, Limited Datacited by 16
  11. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  12. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
38 more
  1. 2022Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertaintycited by 13
  2. 2021Physics-constrained Deep Learning for Robust Inverse ECG Modelingcited by 12
  3. 2021Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Datacited by 12
  4. 2024ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identificationcited by 10
  5. 2021Identifying Physical Law of Hamiltonian Systems via Meta-Learningcited by 8
  6. 2022Symbolic Recovery of Differential Equations: The Identifiability Problemcited by 8
  7. 2023Generalized Neural Closure Models with Interpretabilitycited by 8
  8. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  9. 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
  10. 2023Learning minimal representations of stochastic processes with variational autoencoderscited by 7
  11. 2025MDBench: Benchmarking Data-Driven Methods for Model Discoverycited by 7
  12. 2022Augmenting astrophysical scaling relations with machine learning: application to reducing the Sunyaev-Zeldovich flux-mass scattercited by 6
  13. 2023Discovering stochastic partial differential equations from limited data using variational Bayes inferencecited by 6
  14. 2023A Robust SINDy Approach by Combining Neural Networks and an Integral Formcited by 6
  15. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  16. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  17. 2023Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernelscited by 5
  18. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  19. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  20. 2021Quantum Model-Discoverycited by 4
  21. 2022Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate datacited by 4
  22. 2021Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equationscited by 3
  23. 2023Hierarchical shrinkage Gaussian processes: applications to computer code emulation and dynamical system recoverycited by 3
  24. 2024An invariance constrained deep learning network for PDE discoverycited by 3
  25. 2024On the estimation rate of Bayesian PINN for inverse problemscited by 3
  26. 2024Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systemscited by 3
  27. 2024A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systemscited by 3
  28. 2025Mesh-free sparse identification of nonlinear dynamicscited by 3
  29. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  30. 2021Model discovery in the sparse sampling regimecited by 2
  31. 2021Data-Driven Theory-guided Learning of Partial Differential Equations using SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE)cited by 2
  32. 2021Discovering PDEs from Multiple Experimentscited by 2
  33. 2022Learning deterministic hydrodynamic equations from stochastic active particle dynamicscited by 2
  34. 2024GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equationscited by 2
  35. 2025A Neural Operator-Based Approach to Symbolic Discovery of PDEscited by 2
  36. 2025Evaluating PDE discovery methods for multiscale modeling of biological signalscited by 2
  37. 2025Electron neural closure for turbulent magnetosheath simulations: energy channelscited by 2
  38. 2021Fully differentiable model discoverycited by 1

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