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Unrolled-SINDy: A Stable Explicit Method for Non linear PDE Discovery from Sparsely Sampled Data

Fayad Ali Banna, Antoine Caradot, Eduardo Brandao, Jean-Philippe Colombier, Rémi Emonet, Marc Sebban

TL;DR

Unrolled-SINDy introduces a novel unrolling scheme for explicit PDE discovery that decouples the data sampling interval from the numerical time-step, enabling robust recovery of governing equations from sparsely sampled data. By performing K intermediate updates per observation and averaging the candidate-term dictionaries, the method reduces local truncation error and yields a local truncation error growth of order ${\mathcal{O}}\left(\tfrac{h^p}{K}\right)$ for RK methods of order $p$, with a specific bound ${\mathcal{O}}\left(\tfrac{h^2}{K}\right)$ for Euler. The framework supports closed-form estimation and SGD-based variants, and demonstrates improved performance on 2D reaction–diffusion and Kuramoto–Sivashinsky PDEs, as well as benefiting neural PDE discovery methods like iNeural-SINDy. The results show that unrolling extends the regime where SINDy-based discovery succeeds under large inter-observation gaps and noise, providing practical gains for data-scarce scenarios in physics-informed learning.

Abstract

Identifying from observation data the governing differential equations of a physical dynamics is a key challenge in machine learning. Although approaches based on SINDy have shown great promise in this area, they still fail to address a whole class of real world problems where the data is sparsely sampled in time. In this article, we introduce Unrolled-SINDy, a simple methodology that leverages an unrolling scheme to improve the stability of explicit methods for PDE discovery. By decorrelating the numerical time step size from the sampling rate of the available data, our approach enables the recovery of equation parameters that would not be the minimizers of the original SINDy optimization problem due to large local truncation errors. Our method can be exploited either through an iterative closed-form approach or by a gradient descent scheme. Experiments show the versatility of our method. On both traditional SINDy and state-of-the-art noise-robust iNeuralSINDy, with different numerical schemes (Euler, RK4), our proposed unrolling scheme allows to tackle problems not accessible to non-unrolled methods.

Unrolled-SINDy: A Stable Explicit Method for Non linear PDE Discovery from Sparsely Sampled Data

TL;DR

Unrolled-SINDy introduces a novel unrolling scheme for explicit PDE discovery that decouples the data sampling interval from the numerical time-step, enabling robust recovery of governing equations from sparsely sampled data. By performing K intermediate updates per observation and averaging the candidate-term dictionaries, the method reduces local truncation error and yields a local truncation error growth of order for RK methods of order , with a specific bound for Euler. The framework supports closed-form estimation and SGD-based variants, and demonstrates improved performance on 2D reaction–diffusion and Kuramoto–Sivashinsky PDEs, as well as benefiting neural PDE discovery methods like iNeural-SINDy. The results show that unrolling extends the regime where SINDy-based discovery succeeds under large inter-observation gaps and noise, providing practical gains for data-scarce scenarios in physics-informed learning.

Abstract

Identifying from observation data the governing differential equations of a physical dynamics is a key challenge in machine learning. Although approaches based on SINDy have shown great promise in this area, they still fail to address a whole class of real world problems where the data is sparsely sampled in time. In this article, we introduce Unrolled-SINDy, a simple methodology that leverages an unrolling scheme to improve the stability of explicit methods for PDE discovery. By decorrelating the numerical time step size from the sampling rate of the available data, our approach enables the recovery of equation parameters that would not be the minimizers of the original SINDy optimization problem due to large local truncation errors. Our method can be exploited either through an iterative closed-form approach or by a gradient descent scheme. Experiments show the versatility of our method. On both traditional SINDy and state-of-the-art noise-robust iNeuralSINDy, with different numerical schemes (Euler, RK4), our proposed unrolling scheme allows to tackle problems not accessible to non-unrolled methods.
Paper Structure (55 sections, 2 theorems, 37 equations, 12 figures, 39 tables, 2 algorithms)

This paper contains 55 sections, 2 theorems, 37 equations, 12 figures, 39 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Definitions, Notations and Necessary Background
  3. Euler-SINDy PDE-FIND:
  4. RK4-SINDy RK4-SINDy:
  5. Unrolled SINDy
  6. Unrolling the Time Derivative Numerical Approximation
  7. Analysis of the Truncation Errors
  8. The Unrolled Euler-SINDy Algorithm
  9. Experimental Results
  10. Improvements of SINDy by Unrolling
  11. 2D Reaction-Diffusion PDE system
  12. Experimental setup:
  13. Results:
  14. Kuramoto–Sivashinsky PDE
  15. Experimental setup:
  16. ...and 40 more sections

Key Result

Theorem 1

The local truncation error suffered by the unrolled Euler estimate of Eq. eq:unrolled-factorized (assuming that $\forall j, h_j=h$) is on the order ${\mathcal{O}} \left ( \frac{h^2}{K} \right )$ such that: with the constant $M=\max_{t' \in [t,t+h]}|u"(t')|.$

Figures (12)

  • Figure 1: Left: With the solid black lines showing the predictions using the true coefficients (hence tangent to the trajectory), illustration of how SINDy operates with the Euler method associated with (a) a small time step $h_j$, and (b) where data is sparsely sampled; In the center, we show the benefit of (c) unrolling $K=2$ times, and (d) $K=4$ times each numerical step which leads to a smaller local truncation error; Right: (e) the matrices involved in prediction steps, and (f) the 4-Unrolled Euler-SINDy Algorithm \ref{['algo:Unrolled-SINDy']}.
  • Figure 2: Solutions of the 2D Reaction-Diffusion PDEs (with $h=1$ and $K=25$) simulated from the ground truth and the learned PDEs ($u$ in green and $v$ in red).
  • Figure 3: Solutions of the Kuramoto–Sivashinsky PDE with $h=0.2$. From top to bottom: Euler-SINDy, ground truth, and 10 Unrolled Euler-SINDy.
  • Figure 4: Solution of cubic damped oscillator with noisy data ($\sigma=0.02$) using a) RK4 iNeural-SINDy and b) 2 Unrolled RK4-iNeural-SINDy.
  • Figure 5: Comparison of the reconstruction errors as a function of the execution time (log scale) for gradient descent (SGD) and closed-form SINDy models on the 2-dimentional cubic damped oscillator (where $h=10^{-3}$).
  • ...and 7 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Definition 1
  • Definition 2: Stability of a numerical method for a function $f$