mlsanityData to 7 Oct 2026

Online Weak-form Sparse Identification of Partial Differential Equations

arXiv 2203.03979 ↗First posted 8 Mar 2022math.OC cs.LG stat.MLcited by 13

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Abstract (arXiv)

This paper presents an online algorithm for identification of partial differential equations (PDEs) based on the weak-form sparse identification of nonlinear dynamics algorithm (WSINDy). The algorithm is online in a sense that if performs the identification task by processing solution snapshots that arrive sequentially. The core of the method combines a weak-form discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem. In particular, we do not regularize the ℓ0-pseudo-norm, instead finding that directly applying its proximal operator (which corresponds to a hard thresholding) leads to efficient online system identification from noisy data. We demonstrate the success of the method on the Kuramoto-Sivashinsky equation, the nonlinear wave equation with time-varying wavespeed, and the linear wave equation, in one, two, and three spatial dimensions, respectively. In particular, our examples show that the method is capable of identifying and tracking systems with coefficients that vary abruptly in time, and offers a streaming alternative to problems in higher dimensions.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 21 of its 53 references that are in the map. First posted on arXiv 8 Mar 2022.

Builds on

21 of 53 references.

References · 21

  1. 2017PDE-Net: Learning PDEs from Datacited by 304
  2. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  3. 2018Online Learning: A Comprehensive Surveycited by 203
  4. 2013Non-stationary Stochastic Optimizationcited by 173
  5. 2018Learning Without Mixing: Towards A Sharp Analysis of Linear System Identificationcited by 158
  6. 2020Weak SINDy For Partial Differential Equationscited by 154
  7. 2017Sparse identification of nonlinear dynamics for model predictive control in the low-data limitcited by 139
  8. 2018Learning-based Model Predictive Control for Safe Explorationcited by 116
  9. 2018Data Driven Governing Equations Approximation Using Deep Neural Networkscited by 94
  10. 2017Sparse learning of stochastic dynamic equationscited by 76
  11. 2018On the Convergence of the SINDy Algorithmcited by 49
  12. 2019Data-Driven Deep Learning of Partial Differential Equations in Modal Spacecited by 46
9 more
  1. 2020Learning nonlinear dynamical systems from a single trajectorycited by 46
  2. 2020Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systemscited by 39
  3. 2021Learning Mean-Field Equations from Particle Data Using WSINDycited by 29
  4. 2018Online Learning with Inexact Proximal Online Gradient Descent Algorithmscited by 22
  5. 2019Learning Sparse Dynamical Systems from a Single Sample Trajectorycited by 20
  6. 2019Optimization and Learning with Information Streams: Time-varying Algorithms and Applicationscited by 15
  7. 2020Data-Driven Discovery of Coarse-Grained Equationscited by 14
  8. 2018A Novel Framework for Online Supervised Learning with Feature Selectioncited by 2
  9. 2020Stochastic Hard Thresholding Algorithms for AUC Maximizationcited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 3; 2023: 7; 2024: 1; 2025: 0; 2026: 2.

13 papers.

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  1. 2023Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamicscited by 20
  2. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
  3. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  4. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
  5. 2022Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Populationcited by 11
  6. 2023Coarse-Graining Hamiltonian Systems Using WSINDycited by 11
  7. 2022Random Feature Models for Learning Interacting Dynamical Systemscited by 10
  8. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  9. 2023Streaming Compression of Scientific Data via weak-SINDycited by 6
  10. 2023Discover governing differential equations from evolving systemscited by 5
  11. 2024Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurementscited by 2
  12. 2026Learning effective models from network dynamics data with multiple initial conditions using weak form SINDycited by 2
1 more
  1. 2026Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamicscited by 0