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Dynamics-aware identification of governing equations from sparse and noisy data

This paper demonstrates that Koopman-based upsampling techniques, such as DMD and EDMD, serve as effective dynamics-aware preprocessing steps to denoise and interpolate sparse data, thereby improving the accuracy and stability of identifying governing ODEs and PDEs compared to traditional non-dynamical interpolation methods.

Original authors: Pongpisit Thanasutives, Yoshinobu Kawahara

Published 2026-08-03

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CC BY 4.0

Original authors: Pongpisit Thanasutives, Yoshinobu Kawahara

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but your only clues are a few blurry, shaky photos taken at random moments. You know the suspect is moving, but the photos are so sparse and full of static that you can't tell if they are walking, running, or dancing. In the world of science, this is exactly the challenge of "data-driven discovery." Scientists want to find the hidden mathematical rules (equations) that govern how things move, from the swirling of a storm to the beating of a heart. They have powerful tools like SINDy and PDE-FIND that can look at data and guess these rules. But there's a catch: these tools need to know how fast things are changing at every single moment (the "derivative"). If your data is sparse (few photos) and noisy (blurry), calculating that speed is like trying to guess the speed of a race car by looking at two photos taken ten minutes apart; the math breaks down, and the detective gets the wrong answer.

This is where the story gets interesting. The researchers in this paper asked a simple question: "What if we could fill in the missing photos before we try to solve the mystery?" Instead of just guessing the speed from the blurry snapshots, they used a clever trick called "Koopman-based upsampling." Think of it like a smart, time-traveling photo editor. It studies the few photos you have, learns the underlying "rhythm" or "dance" of the system, and then generates high-quality, clear intermediate photos to fill in the gaps. Once the detective has a smooth, continuous movie instead of a flickering slideshow, they can finally calculate the speed accurately and discover the true rules of the game.

The paper, titled "Dynamics-aware identification of governing equations from sparse and noisy data," puts this idea to the test. The authors, Pongpisit Thanasutives and Yoshinobu Kawahara, ran a massive simulation experiment to see if this "smart photo editing" actually helps scientists find the right equations. They didn't just guess; they tested it on two famous chaotic systems (Lorenz–63 and Van der Pol) and three complex wave systems (Burgers, Fisher–KPP, and linear advection–diffusion). They simulated data that was both sparse (missing many time steps) and noisy (full of static), just like real-world sensor data often is.

The results were a mix of "huge wins" and "it depends." For the simpler, chaotic systems (the ODEs), the smart editing worked like a charm. Specifically, a method called "Polynomial EDMD" acted like a magic lens. It reduced the error in the calculated numbers by a massive margin. For the Van der Pol system, the error dropped from a messy 0.107 down to a crisp 0.015. It was as if the detective went from guessing the suspect's speed with a shaky hand to measuring it with a laser. The paper suggests that when the underlying rules are polynomial (made of simple math powers), this method is the clear winner.

However, for the more complex wave systems (the PDEs), the story was a bit more nuanced. The "smart editing" didn't work equally well for every system. For the Burgers equation (which models shock waves) and the advection–diffusion system (which models how smoke spreads), the method was fantastic, cutting errors down significantly. But for the Fisher–KPP system (which models how populations spread), the "raw" data without any editing was actually almost as good as the edited version. The paper suggests that the benefit depends heavily on the specific "dance" the system is doing; if the system's rhythm fits the editor's style, it works wonders. If not, the extra editing might not add much value.

Crucially, the authors compared their "smart, physics-aware" editor against standard, boring editors like simple linear interpolation (drawing straight lines between dots) and smoothing splines (curving lines to fit the dots). The paper found that while the boring editors were better than doing nothing, the "smart" editors that understood the system's dynamics were consistently superior. They didn't just fill in the gaps; they filled them in with the right kind of movement.

In the end, the paper doesn't claim to have solved the mystery of all data. It suggests that this "Koopman-based upsampling" is a powerful new tool in the detective's kit, but it works best when the tool matches the case. If the data is sparse and noisy, and the system has a rhythm the tool can learn, then filling in the gaps before doing the math can turn a blurry, confusing mess into a clear, solvable equation. It's a reminder that sometimes, to find the future, you first have to reconstruct the missing moments of the past.

Technical Summary: Dynamics-aware Identification of Governing Equations from Sparse and Noisy Data

Problem Statement

Data-driven discovery of governing equations, such as Sparse Identification of Nonlinear Dynamics (SINDy) for ODEs and PDE functional identification (PDE-FIND) for PDEs, relies on regressing estimated time derivatives against a library of candidate functions. A critical limitation in these methods is the reliability of derivative estimates when the input data is both temporally sparse and noisy. Finite-difference derivatives computed from such data often become the primary source of error, leading to incorrect support recovery (identifying the wrong terms) or biased coefficient estimation, even when the correct candidate terms exist in the library. While robust regression variants exist, this paper investigates a preprocessing approach to stabilize the derivative estimation step itself.

Methodology

The authors propose a Koopman-based upsampling strategy as a dynamics-aware preprocessing step. Instead of differentiating sparse, noisy snapshots directly, the method fits a finite-dimensional approximation of the Koopman operator to the observed data, interpolates intermediate snapshots within the observed time window, and then performs derivative estimation and sparse regression on the reconstructed, denser trajectory.

The core components of the methodology include:

Koopman Approximations via DMD Variants:
- DMD (Dynamic Mode Decomposition): Fits a linear map between consecutive state snapshots.
- EDMD (Extended DMD): Lifts the state to a richer observable space (e.g., polynomials or Radial Basis Functions) to capture nonlinear dynamics via a linear map on observables.
- Optimized DMD (optDMD): Fits a continuous-time low-rank exponential model directly to the snapshots.
- POD-EDMD: For high-dimensional PDEs, Proper Orthogonal Decomposition (POD) is first used to project fields onto a low-rank subspace, followed by EDMD on the latent coordinates.
Interpolation Strategy:
- The primary strategy is local-reset interpolation. The learned model advances the state from an observed snapshot to intermediate points, but the process is reset at each subsequent observed snapshot. This prevents the accumulation of model error (drift) common in long-horizon forecasting.
- Alternative strategies (global rollout, residual-corrected rollout, two-sided interpolation) were tested in ablation studies but found to be less robust or only marginally beneficial in specific stress conditions.
Sparse Regression Pipeline:
- After upsampling, temporal (and spatial for PDEs) derivatives are estimated using central finite differences on the reconstructed data.
- STLSQ (Sequential Thresholded Least Squares) is used to identify the sparse coefficients.
- Evaluation: The study employs an "oracle" diagnostic where the true support and coefficients are known to select the optimal threshold from a grid, maximizing a practical score ( $S$ ) that balances F1 support recovery and coefficient error. A separate non-oracle experiment uses Extended Bayesian Information Criterion (EBIC) and Pareto optimization for model selection without ground truth.

Key Contributions and Results

1. Ordinary Differential Equations (ODEs)

The benchmark tested Lorenz–63 and Van der Pol systems under various noise levels and sparse factors.

Polynomial EDMD emerged as the strongest preprocessor. It significantly reduced median coefficient error (e.g., from 0.911 to 0.526 for Lorenz–63; from 0.107 to 0.015 for Van der Pol) and improved the practical score.
The improvement is attributed to the alignment between the polynomial EDMD dictionary and the polynomial structure of the target ODEs.
Comparison: While linear interpolation and tuned smoothing splines improved over the raw baseline, Polynomial EDMD consistently achieved the best coefficient accuracy and practical scores.

2. Partial Differential Equations (PDEs)

The benchmark tested periodic Burgers, Fisher–Kolmogorov–Petrovskii–Piskunov (Fisher–KPP), and linear advection–diffusion systems.

System-Dependent Gains: Unlike the ODE results, PDE improvements were not uniform.
- Burgers & Advection-Diffusion: Low-rank DMD-assisted reconstructions (optDMD and POD-EDMD-RBF) substantially improved coefficient accuracy and practical scores.
- Fisher–KPP (Standard IC): The raw baseline remained highly competitive; assisted methods offered only modest gains.
- Fisher–KPP (Steep Front IC): When the initial condition contained sharper spatial structures, POD-EDMD-RBF provided significant gains, suggesting the benefit depends on the information content of the transient.
POD-EDMD-RBF was identified as the most reliable PDE preprocessor across the full grid, outperforming linear and spline interpolation in aggregate metrics.

3. Model Selection Stability

The study demonstrated that DMD-assisted upsampling stabilizes Pareto-based non-oracle support-size selection. In experiments using EBIC and elbow detection without access to ground truth:

The raw baseline often failed to recover the correct support size for complex cases (e.g., Fisher–KPP with steep fronts).
The DMD-assisted selector successfully recovered the exact support and correct support size across all seeds and target equations.

Significance and Claims

The paper claims that Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step rather than a universal solver. Its effectiveness is contingent on the match between the observable representation (dictionary), the low-rank structure of the data, and the sampling regime.

For ODEs: The method provides clear gains in coefficient accuracy when the dictionary (e.g., polynomials) aligns with the system's nonlinearity.
For PDEs: The method acts as a joint temporal interpolator and spatial denoiser (via POD), proving most effective when the field evolution is compatible with the learned low-rank representation.
Comparison to Alternatives: The study concludes that these dynamics-aware preprocessors generally outperform classical non-dynamical alternatives (linear interpolation and smoothing splines), particularly in reducing coefficient bias and stabilizing model selection.

The authors emphasize that while the "oracle" threshold selection used for benchmarking is not deployable in real-world scenarios, the non-oracle EBIC/Pareto experiments confirm that upsampling makes the complexity–fit trade-off easier to identify, thereby aiding practical equation discovery. The work isolates specific regimes where this preprocessing improves equation recovery, offering a practical tool for handling sparse and noisy measurements in scientific modeling.