Mesh-free sparse identification of nonlinear dynamics

2025-05-21

AI Summary

Conventional dynamical modeling methods rely heavily on structured spatiotemporal grids and high-fidelity data, rendering them unsuitable for real-world scenarios involving unstructured sensor placement and irregular, asynchronous time sampling. Method: This paper introduces Mesh-free SINDy—the first grid-free sparse identification framework—integrating neural network-based function approximation, automatic differentiation, and sparse regression to directly infer nonlinear PDE governing equations from discrete, asynchronous, and sparse observations without explicit spatial or temporal discretization. Contribution/Results: Evaluated on four canonical PDEs (including Burgers’ equation), Mesh-free SINDy achieves robust equation recovery under strong noise (up to 75% additive noise), successfully identifies dynamics from as few as 100 time-series samples with only 1% noise, and completes training in under one minute. The framework significantly enhances the robustness and practicality of sparse identification in low signal-to-noise ratio, small-data, and irregular-observation regimes.

Technology Category

• Machine Learning: Learning with Manifolds
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Identifying the governing equations of a dynamical system is one of the most important tasks for scientific modeling. However, this procedure often requires high-quality spatio-temporal data uniformly sampled on structured grids. In this paper, we propose mesh-free SINDy, a novel algorithm which leverages the power of neural network approximation as well as auto-differentiation to identify governing equations from arbitrary sensor placements and non-uniform temporal data sampling. We show that mesh-free SINDy is robust to high noise levels and limited data while remaining computationally efficient. In our implementation, the training procedure is straight-forward and nearly free of hyperparameter tuning, making mesh-free SINDy widely applicable to many scientific and engineering problems. In the experiments, we demonstrate its effectiveness on a series of PDEs including the Burgers' equation, the heat equation, the Korteweg-De Vries equation and the 2D advection-diffusion equation. We conduct detailed numerical experiments on all datasets, varying the noise levels and number of samples, and we also compare our approach to previous state-of-the-art methods. It is noteworthy that, even in high-noise and low-data scenarios, mesh-free SINDy demonstrates robust PDE discovery, achieving successful identification with up to 75% noise for the Burgers' equation using 5,000 samples and with as few as 100 samples and 1% noise. All of this is achieved within a training time of under one minute.

Problem

• Identifies nonlinear dynamics from arbitrary sensor data
• Robust to high noise and limited data samples
• Computationally efficient with minimal hyperparameter tuning

Innovation

• Mesh-free SINDy algorithm for nonlinear dynamics
• Neural network and auto-differentiation for arbitrary data
• Robust to high noise and limited data efficiently