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Robust Sparse Identification of Nonlinear Dynamics via Least Trimmed Squares

This paper proposes a robust SINDy framework called ILTS-SINDy that combines Iterative Least Trimmed Squares for outlier filtering with Sequentially Thresholded Least Squares for sparse regression, demonstrating superior performance in identifying nonlinear dynamics from datasets with up to 20% corrupted observations.

Original authors: Fabio Amaral, Geovani N. Grapiglia, Cassio M. Oishi

Published 2026-06-30✓ Author reviewed ⓘ

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabio Amaral, Geovani N. Grapiglia, Cassio M. Oishi

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 by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out the secret recipe for a delicious soup by tasting it at different moments in time. You want to write down the exact list of ingredients (the "governing equations") that create the flavor.

In the world of science, this is what SINDy (Sparse Identification of Nonlinear Dynamics) does. It looks at data from a system (like weather patterns, animal populations, or disease spread) and tries to find the simple mathematical rules that explain how that system changes over time.

However, real-world data is messy. Sometimes the thermometer breaks, a sensor glitches, or someone accidentally spills a drop of soup on the measuring cup. In data science, these are called outliers or noise. If you try to figure out the recipe using a spoonful of soup that has a piece of glass in it, your final recipe will be wrong.

The Problem with Old Methods

The paper explains that standard ways of finding these recipes are very sensitive to "bad data."

Standard SINDy: It tries to use every data point, even the bad ones. If the data is noisy, the math gets confused, and it invents fake ingredients (like "magic dust") that don't actually exist.
Other "Robust" Methods: Some newer methods try to fix this by averaging many different guesses or by trying to remove bad data while they are calculating the recipe. The authors argue these methods are either too slow, too complicated, or they still get confused because they are trying to do two hard things at once.

The New Solution: ILTS-SINDy

The authors propose a new, two-step pipeline called ILTS-SINDy. They describe it as a "filter-then-sparsify" approach. Think of it like a two-stage cooking process:

Stage 1: The "Smart Strainer" (ILTS)

Before you even start writing down the recipe, you need to clean your ingredients.

The Analogy: Imagine you have a bucket of water with some rocks and leaves in it. Instead of trying to drink the water while the rocks are still in it, you pour it through a smart strainer.
How it works: The method uses an algorithm called Iterative Least Trimmed Squares (ILTS). It looks at all the data points and asks, "Which ones fit the pattern best?" It keeps the "good" data points (the clear water) and throws away the "bad" ones (the rocks and leaves) that don't make sense. It does this over and over again until it is sure it has only the cleanest, most reliable data left.
The Result: You now have a clean dataset, free from the weird glitches and outliers.

Stage 2: The "Minimalist Chef" (STLS)

Now that you have clean ingredients, you can figure out the recipe.

The Analogy: A minimalist chef wants to use the fewest ingredients possible to make the soup taste right. They don't want to add salt, pepper, garlic, and basil if only salt and pepper are needed.
How it works: The method uses Sequentially Thresholded Least Squares (STLS). It looks at the clean data and tries to find the simplest mathematical equation that fits. It starts by considering all possible ingredients, then systematically removes the ones that aren't necessary until only the essential "sparse" (simple) rules remain.

Why This is a Big Deal

The authors tested this new method on three famous "soup recipes" (mathematical models for disease spread, chaotic weather, and predator-prey animal populations). They intentionally added "rotten ingredients" (outliers) to the data to see how well the methods could handle it.

The Results:
- Standard SINDy failed miserably when the data was dirty. It couldn't find the right recipe.
- Other Robust Methods did okay, but they still struggled when the data was very messy.
- ILTS-SINDy was the champion. Even when up to 20% of the data was corrupted (meaning 1 out of every 5 data points was a lie or a glitch), ILTS-SINDy could still find the exact, correct mathematical recipe.

The Bottom Line

The paper claims that by separating the job of cleaning the data from the job of finding the rules, the new method is much better at ignoring noise and finding the true underlying laws of nature. It's like having a chef who refuses to taste a dish until the kitchen is spotless, ensuring the final recipe is perfect.

Technical Summary: Robust Sparse Identification of Nonlinear Dynamics via Least Trimmed Squares

Problem Statement
The Sparse Identification of Nonlinear Dynamics (SINDy) framework is a prominent data-driven method for discovering governing differential equations from observational data. However, standard SINDy implementations are highly sensitive to measurement noise and, critically, to outliers. Since SINDy relies on finite-difference approximations of time derivatives, corrupted data points can severely distort derivative estimates, leading to biased regression problems and the identification of spurious model terms. While existing robust variants—such as Trimmed SINDy, Weak SINDy (WSINDy), and Ensemble SINDy (E-SINDy)—have been proposed, they face limitations. Trimmed approaches often couple outlier detection with sparse regression in a single non-convex optimization, potentially failing to fully exclude corrupted samples. WSINDy reduces noise sensitivity but introduces complexity in selecting test functions and integration domains. E-SINDy improves stability through resampling but can be computationally expensive and may still inherit biases from corrupted subsamples.

Methodology: The ILTS-SINDy Pipeline
The authors propose ILTS-SINDy, a robust identification pipeline that decouples outlier filtering from sparse regression into two sequential stages. This "filter-then-sparsify" strategy contrasts with methods that perform these tasks jointly.

Stage 1: Iterative Least Trimmed Squares (ILTS) Filtering
The first stage addresses data contamination by identifying and removing outliers before model fitting. The method employs the Iterative Least Trimmed Squares (ILTS) algorithm.
- Mechanism: ILTS iteratively minimizes the sum of the smallest squared residuals. It alternates between estimating regression coefficients on a subset of data and updating this subset to retain only the $p$ observations with the smallest residuals.
- Outcome: This process progressively concentrates on the most consistent data points, effectively mitigating the influence of outliers without requiring prior knowledge of their locations. The output is a "cleaned" dataset (a reduced matrix $\hat{A}$ and vector $\hat{b}$ ) containing only the estimated inliers.
- Parameter Selection: The number of inliers $p$ is set based on the known outlier fraction $q$ . The authors note that due to the use of central finite differences, a single outlier can contaminate up to three derivative estimates (the point itself and its neighbors). Consequently, they set $p = (m+1) - \lfloor 3q(m+1) \rfloor$ .
Stage 2: Sequentially Thresholded Least Squares (STLS)
The second stage performs sparse model identification on the cleaned dataset.
- Mechanism: Standard STLS is applied to the filtered data. It solves a least-squares problem using the full candidate library and iteratively enforces sparsity by thresholding small coefficients to zero. The regression is re-solved using only the remaining active terms until convergence.
- Result: This yields a parsimonious representation of the governing equations ( $\dot{x}_i(t) = \sum \alpha_{i,j} \theta_j(x(t))$ ) that is robust to the initial data corruption.

Key Contributions

Decoupled Framework: The primary contribution is the structural separation of outlier detection (via ILTS) and sparse regression (via STLS). This avoids the trade-offs inherent in joint optimization approaches where corrupted samples might be misclassified as inliers during the sparse estimation phase.
Robustness to Severe Corruption: The method is designed to handle datasets with significant outlier contamination (up to 20% in the experiments) while maintaining the ability to recover the true governing equations.
Algorithmic Integration: The paper formalizes the integration of ILTS (Algorithm 2) and STLS (Algorithm 1) into a unified pipeline (Algorithm 3) specifically for SINDy applications.

Numerical Results
The authors evaluated ILTS-SINDy against standard SINDy (with STLS) and E-SINDy using synthetic datasets from three benchmark dynamical systems: the SIR model, the Lorenz system, and the Lotka–Volterra system. Experiments involved varying levels of additive Gaussian noise and outlier percentages ( $q$ ) ranging from 2.5% to 20%.

Support Recovery Accuracy: ILTS-SINDy consistently achieved the highest mean support recovery accuracy across all systems and conditions. While standard SINDy performance degraded significantly with increased corruption, and E-SINDy showed moderate robustness, ILTS-SINDy maintained accuracy values close to 1.0 even under severe contamination.
Exact Recovery Rate: A critical metric was the exact recovery of the true sparsity pattern (the correct set of active terms). Standard SINDy failed to achieve exact recovery in most corrupted scenarios. E-SINDy showed high support accuracy but low exact recovery rates, suggesting it often reconstructed trajectories correctly but identified the wrong model structure. In contrast, ILTS-SINDy demonstrated superior exact recovery rates, maintaining near-perfect recovery for the SIR system and strong performance for Lorenz and Lotka–Volterra systems even at high outlier levels.

Significance and Claims
The paper claims that ILTS-SINDy offers a significant improvement over existing robust SINDy variants, particularly in regimes of strong data contamination. By filtering outliers prior to sparse regression, the method produces a cleaner regression problem, leading to more reliable identification of the underlying governing equations. The authors assert that this approach not only improves predictive accuracy but, more importantly, ensures the correct structural discovery of the dynamical system.

The work is modest regarding its current scope, noting that it assumes exact knowledge of the outlier percentage $q$ to determine the number of inliers $p$ . Future work is identified as incorporating an adaptive procedure to estimate $q$ data-driven, extending the framework to SINDy with control (external inputs), and validating the approach on real-world datasets. The paper does not claim universal applicability to all dynamical systems without further validation but highlights its efficacy on the tested synthetic benchmarks.