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

Yuhuang Meng, Yue Qiu

TL;DR

This work addresses robust sparse discovery of differential equations from noisy data by introducing GP-SINDy, which uses Gaussian process surrogates to denoise observations and quantify uncertainty in state and derivative estimates, and MFGP-SINDy, which fuses low- and high-fidelity data via a multi-fidelity Gaussian process kernel. The methods yield a sparse representation of the dynamics by weighting the derivative regression with posterior uncertainty and by exploiting MF information to reduce the need for expensive high-fidelity data. Across Lorenz, Burgers, and KdV benchmarks, GP-SINDy demonstrates strong noise-robustness, while MFGP-SINDy achieves superior accuracy with less HF data, highlighting data efficiency and uncertainty-aware discovery. These approaches enhance the practical applicability of data-driven differential equation discovery in situations with limited or noisy measurements and varying data fidelities.

Abstract

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.

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

TL;DR

This work addresses robust sparse discovery of differential equations from noisy data by introducing GP-SINDy, which uses Gaussian process surrogates to denoise observations and quantify uncertainty in state and derivative estimates, and MFGP-SINDy, which fuses low- and high-fidelity data via a multi-fidelity Gaussian process kernel. The methods yield a sparse representation of the dynamics by weighting the derivative regression with posterior uncertainty and by exploiting MF information to reduce the need for expensive high-fidelity data. Across Lorenz, Burgers, and KdV benchmarks, GP-SINDy demonstrates strong noise-robustness, while MFGP-SINDy achieves superior accuracy with less HF data, highlighting data efficiency and uncertainty-aware discovery. These approaches enhance the practical applicability of data-driven differential equation discovery in situations with limited or noisy measurements and varying data fidelities.

Abstract

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.
Paper Structure (23 sections, 2 theorems, 53 equations, 8 figures, 11 tables, 4 algorithms)

This paper contains 23 sections, 2 theorems, 53 equations, 8 figures, 11 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Sparse identification using Gaussian process
  4. Gaussian process regression
  5. Inferring the state variables
  6. Inferring the partial derivatives of state variables
  7. Training the hyperparameters in GP
  8. GP-SINDy for single-fidelity data
  9. MFGP-SINDy for multi-fidelity data
  10. MFGP construction
  11. MFGP-SINDy
  12. Numerical experiments
  13. Lorenz system
  14. Burgers' equation
  15. GP-SINDy for SF data
  16. ...and 8 more sections

Key Result

Lemma 1

For the least squares problem arising from Equation equ:wls-noise-pre with $\boldsymbol{\Sigma}_i = \sigma^2\mathbf{I}$, the best linear unbiased estimator (BLUE) is given by $\hat{\mathbf{C}_{i}} = (\boldsymbol{\Phi}^T\boldsymbol{\Phi})^{-1}\boldsymbol{\Phi}^T\overline{\dot{\mathbf{U}}}_i$.

Figures (8)

  • Figure 1: The ground truth and noisy data with $\sigma_\text{NR}=0.1$.
  • Figure 2: Comparison between the ground truth, training data, and states prediction obtained from GPR.
  • Figure 3: The prediction and uncertainty approximation of derivatives.
  • Figure 4: The visualization of discovered dynamic systems, and these figures correspond to Table \ref{['tab:lorenz-2']}, where all initial conditions are set to [-8, 7, 27].
  • Figure 5: The HF data and LF data (before interpolation) in Burgers' equation \ref{['equ:burgers']}.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Remark 1
  • Lemma 1: Gauss-Markov Theorem hansen2022modern
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4