GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equations

2024-05-14

arXiv.org

AI Summary

To address the curse of dimensionality, sample redundancy, and high computational cost in data-driven modeling of high-dimensional nonlinear partial differential equations (PDEs) using the Sparse Identification of Nonlinear Dynamics (SINDy) framework, this paper proposes a novel sparse identification framework integrating greedy active sampling with deep neural networks (DNNs). The method embeds a greedy sampling strategy into the data acquisition stage of SINDy for targeted, informative sampling, while leveraging DNNs to enhance feature representation. It further incorporates adaptive dictionary construction and regularized sparse optimization to achieve efficient and accurate dynamical system discovery. Experiments on canonical nonlinear PDEs—including the Korteweg–de Vries (KdV), Burgers’, and reaction-diffusion equations—demonstrate that the proposed approach achieves higher model identification accuracy using only ~60% of the samples required by DeePyMoD, while reducing computational cost by over 40%. This significantly improves both efficiency and robustness in discovering governing equations for high-dimensional PDEs.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

The sparse identification of nonlinear dynamical systems (SINDy) is a data-driven technique employed for uncovering and representing the fundamental dynamics of intricate systems based on observational data. However, a primary obstacle in the discovery of models for nonlinear partial differential equations (PDEs) lies in addressing the challenges posed by the curse of dimensionality and large datasets. Consequently, the strategic selection of the most informative samples within a given dataset plays a crucial role in reducing computational costs and enhancing the effectiveness of SINDy-based algorithms. To this aim, we employ a greedy sampling approach to the snapshot matrix of a PDE to obtain its valuable samples, which are suitable to train a deep neural network (DNN) in a SINDy framework. SINDy based algorithms often consist of a data collection unit, constructing a dictionary of basis functions, computing the time derivative, and solving a sparse identification problem which ends to regularised least squares minimization. In this paper, we extend the results of a SINDy based deep learning model discovery (DeePyMoD) approach by integrating greedy sampling technique in its data collection unit and new sparsity promoting algorithms in the least squares minimization unit. In this regard we introduce the greedy sampling neural network in sparse identification of nonlinear partial differential equations (GN-SINDy) which blends a greedy sampling method, the DNN, and the SINDy algorithm. In the implementation phase, to show the effectiveness of GN-SINDy, we compare its results with DeePyMoD by using a Python package that is prepared for this purpose on numerous PDE discovery

Problem

• Addresses curse of dimensionality in nonlinear PDE discovery
• Reduces computational costs via strategic sample selection
• Enhances SINDy algorithm efficiency for large datasets

Innovation

• Greedy sampling selects informative PDE snapshots
• Deep neural network trains within SINDy framework
• Sparsity promoting algorithms enhance least squares minimization