# Sparse discovery of differential equations based on multi-fidelity Gaussian process

arXiv: https://arxiv.org/abs/2401.11825 · HTML: https://mlsanity.com/paper/2401.11825/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 22 Jan 2024.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [When Gaussian Process Meets Big Data: A Review of Scalable GPs](https://mlsanity.com/paper/1807.01065/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Koopman operator learning using invertible neural networks](https://mlsanity.com/paper/2306.17396/) (2023)
- [A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study](https://mlsanity.com/paper/2201.12683/) (2022)

## Built on by (most cited first, 5 of 5)

- [Generative Discovery of Partial Differential Equations by Learning from Math Handbooks](https://mlsanity.com/paper/2505.05869/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements](https://mlsanity.com/paper/2606.15690/) (2026)
- [Joint discovery of governing partial differential equations from multi-source datasets by competitive optimization](https://mlsanity.com/paper/2606.30699/) (2026)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)

## Abstract (arXiv)

Sparse identification of differential equations aims to compute the analytic expressions from the observed data explicitly. However, there exist two primary challenges. Firstly, it exhibits sensitivity to the noise in the observed data, particularly for the derivatives computations. Secondly, existing literature predominantly concentrates on single-fidelity (SF) data, which imposes limitations on its applicability due to the computational cost. In this paper, we present two novel approaches to address these problems from the view of uncertainty quantification. We construct a surrogate model employing the Gaussian process regression (GPR) to mitigate the effect of noise in the observed data, quantify its uncertainty, and ultimately recover the equations accurately. Subsequently, we exploit the multi-fidelity Gaussian processes (MFGP) to address scenarios involving multi-fidelity (MF), sparse, and noisy observed data. We demonstrate the robustness and effectiveness of our methodologies through several numerical experiments.