# Online Weak-form Sparse Identification of Partial Differential Equations

arXiv: https://arxiv.org/abs/2203.03979 · HTML: https://mlsanity.com/paper/2203.03979/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 53 references that are in the map.
- First posted on arXiv 8 Mar 2022.

## Builds on (most cited first, 21 of 21)

- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Online Learning: A Comprehensive Survey](https://mlsanity.com/paper/1802.02871/) (2018)
- [Non-stationary Stochastic Optimization](https://mlsanity.com/paper/1307.5449/) (2013)
- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Learning-based Model Predictive Control for Safe Exploration](https://mlsanity.com/paper/1803.08287/) (2018)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [Learning nonlinear dynamical systems from a single trajectory](https://mlsanity.com/paper/2004.14681/) (2020)
- [Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2002.08538/) (2020)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Online Learning with Inexact Proximal Online Gradient Descent Algorithms](https://mlsanity.com/paper/1806.00202/) (2018)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [Optimization and Learning with Information Streams: Time-varying Algorithms and Applications](https://mlsanity.com/paper/1910.08123/) (2019)
- [Data-Driven Discovery of Coarse-Grained Equations](https://mlsanity.com/paper/2002.00790/) (2020)
- [A Novel Framework for Online Supervised Learning with Feature Selection](https://mlsanity.com/paper/1803.11521/) (2018)
- [Stochastic Hard Thresholding Algorithms for AUC Maximization](https://mlsanity.com/paper/2011.02396/) (2020)

## Built on by (most cited first, 13 of 13)

- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Learning Anisotropic Interaction Rules from Individual Trajectories in a Heterogeneous Cellular Population](https://mlsanity.com/paper/2204.14141/) (2022)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [Random Feature Models for Learning Interacting Dynamical Systems](https://mlsanity.com/paper/2212.05591/) (2022)
- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Streaming Compression of Scientific Data via weak-SINDy](https://mlsanity.com/paper/2308.14962/) (2023)
- [Discover governing differential equations from evolving systems](https://mlsanity.com/paper/2301.07863/) (2023)
- [Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurements](https://mlsanity.com/paper/2409.20510/) (2024)
- [Learning effective models from network dynamics data with multiple initial conditions using weak form SINDy](https://mlsanity.com/paper/2605.30432/) (2026)
- [Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamics](https://mlsanity.com/paper/2609.09434/) (2026)

## 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.