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Operator Learning for Robust Discovery of Fractional-Order Dynamical Systems from Noisy Data

This paper proposes a robust framework for discovering governing equations of fractional-order dynamical systems from noisy data by integrating a Deep Operator Network (DeepONet) for accurate fractional derivative estimation with ensemble Sparse Identification of Nonlinear Dynamics (SINDy) for sparse regression, demonstrating superior accuracy and generalization compared to classical finite difference methods and standard neural networks.

Original authors: Yones Yousefpur Azar, Hossein Kheiri, Hamidreza Marasi

Published 2026-09-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yones Yousefpur Azar, Hossein Kheiri, Hamidreza Marasi

Original paper licensed under CC BY 4.0 (https://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

Science often seeks to write the rulebook for how the world moves, from the flow of blood in a vein to the vibration of a bridge. For many systems, these rules are written as equations that describe how things change over time. However, some systems have a memory; their current state depends not just on what is happening right now, but on everything that happened before. To describe this kind of behavior, scientists use a special type of math called fractional calculus. Unlike standard equations that look at immediate changes, these fractional equations look at the entire history of a system. This makes them incredibly powerful for modeling real-world phenomena like the way materials stretch and relax, or how diseases spread through a population. But this power comes with a cost: because these equations rely on the entire past, they are notoriously difficult to solve when the data is messy. In the real world, measurements are never perfect; they are always tainted by small errors and noise. When scientists try to use standard math tools to figure out the rules of these memory-based systems from noisy data, the small errors get amplified, often leading to completely wrong conclusions.

A team of researchers at the University of Tabriz has developed a new way to solve this problem, allowing them to discover the hidden rules of these complex systems even when the data is imperfect. Their approach, described in a recent study, combines two distinct ideas: a type of artificial intelligence designed to understand patterns in data, and a statistical method that finds the simplest possible equations that fit that data. The researchers realized that the main bottleneck was not finding the equation itself, but accurately calculating the "memory" part of the equation from noisy numbers. To fix this, they trained a neural network to act as a specialized translator. This network learned, through millions of simulated examples, how to look at a messy, noisy history of a system and instantly produce a clean, accurate estimate of its fractional rate of change. Once this translator was trained, it could be used on real, noisy measurements to provide the clean data needed for the second step: identifying the governing equation.

The researchers tested this two-step process on several different systems, including simple linear models and more complex, non-linear ones that behave chaotically. They compared their new method against the traditional mathematical tools used for decades and against a standard type of neural network. The results were striking. In tests with linear systems, the traditional method failed almost entirely, producing errors so large that the resulting equations were useless. The standard neural network performed better but still struggled when the data changed slightly from what it had seen before. The new method, however, remained robust. It successfully identified the correct equations with high precision, even when the input data contained significant noise. The researchers found that for simpler systems, the accuracy of the derivative estimation was the key; once they cleaned up the data using their new translator, the equation-finding step worked perfectly.

However, the story became more nuanced when they moved to complex, non-linear systems. Here, even when the researchers provided the perfect, noise-free derivative data, the equation-finding step still struggled to find the right answer. This revealed a critical insight: for these more complicated systems, the problem shifts. It is no longer just about cleaning the data; the difficulty lies in the mathematical challenge of picking the right equation from a vast library of possibilities when the data is noisy. The researchers suggest that while their new method solves the data-cleaning problem effectively, future breakthroughs will likely come from improving the equation-finding tools themselves, rather than just making better data cleaners.

The study also explored how well this method works when faced with situations it has never seen before, such as a system starting from a completely different state or running for a longer time than the training data. In these tests, the new method held up significantly better than the standard neural network, which tended to fail when the conditions shifted. This suggests that the new approach learns the underlying structure of the system's behavior rather than just memorizing specific data points. The researchers note that this method is particularly useful in engineering and biology, where scientists often know the general family of equations that describe a type of system but need to find the specific rules for a new, individual experiment. By training the translator once on simulated data, it can be reused for many different real-world experiments without needing to be retrained, saving time and computational power.

Ultimately, this work demonstrates that the biggest hurdle in understanding memory-based systems is not the complexity of the equations themselves, but the reliability of the data used to find them. By using a specialized artificial intelligence to clean the data before the math begins, the researchers have shown a clear path to more accurate discovery. While the method still faces challenges with the most complex, chaotic systems, it represents a significant step forward. It suggests that by separating the task of cleaning the data from the task of finding the rules, scientists can build more reliable models of the world, turning noisy, imperfect observations into clear, understandable laws of nature.

Technical Summary: Operator Learning for Robust Discovery of Fractional-Order Dynamical Systems from Noisy Data

Problem Statement
The identification of governing equations for fractional-order dynamical systems from noisy experimental data faces a critical bottleneck: the estimation of fractional derivatives. Unlike integer-order derivatives, fractional derivatives (specifically the Caputo derivative) are nonlocal operators that depend on the entire history of the solution. This property makes numerical differentiation highly sensitive to measurement noise; even small perturbations in observed trajectories are amplified during the accumulation of historical data, leading to inaccurate derivative estimates. Since sparse system identification methods, such as SINDy (Sparse Identification of Nonlinear Dynamics), rely heavily on the quality of these derivative inputs, noise-induced errors in derivative estimation propagate directly into the discovery process, often resulting in incorrect model structures or unreliable coefficients. While recent attempts have adapted SINDy to fractional systems, they largely rely on classical numerical approximations (e.g., Fractional Finite Differences, FFD) that remain unstable in noisy environments.

Methodology
The authors propose a hybrid, two-stage framework that decouples fractional derivative estimation from governing equation discovery. This approach separates the learning of the differentiation operator from the sparse regression task to improve robustness and reusability.

Offline Operator Learning (DeepONet):
- A Deep Operator Network (DeepONet) is trained offline on synthetic trajectories generated from a known family of fractional-order systems.
- The network learns the mapping $G: (x(t), t) \mapsto D^\alpha x(t)$ , where the input is the noisy state trajectory and the temporal coordinate, and the supervision target is the exact fractional derivative computed from the governing equations.
- The architecture consists of a branch network (processing the state history) and a trunk network (encoding time), combined via an inner product.
- Once trained, this network acts as a reusable, data-driven surrogate for the Caputo derivative. It processes noisy measurements to produce denoised derivative estimates. Unlike FFD, which has $O(N_t^2)$ complexity, the DeepONet inference scales linearly $O(N_t)$ .
Online Sparse Identification (Ensemble SINDy):
- The denoised derivative estimates generated by the DeepONet are fed into an Ensemble SINDy algorithm.
- A candidate library of nonlinear functions (polynomials) is constructed from the measured states.
- The governing equations are identified by solving a sparse regression problem ( $\hat{D}^\alpha X = \Theta(X)\Xi$ ) using Sequential Thresholded Least Squares (STLS) across multiple bootstrap samples. The final coefficients are aggregated based on their frequency of selection to ensure stability.

Key Contributions

Hybrid Framework: The paper introduces a novel pipeline combining pre-trained DeepONets with Ensemble SINDy to robustly discover fractional-order equations from noisy data.
Decoupled Strategy: By separating derivative estimation from equation discovery, the method allows the operator learning component to be optimized specifically for noise robustness, while preserving the interpretability of the final sparse model.
Comprehensive Benchmarking: The study provides a rigorous comparison against classical Fractional Finite Differences (FFD) and a conventional Multilayer Perceptron (MLP) baseline across linear and nonlinear systems, various noise levels, and different fractional orders.
Generalization Analysis: The work evaluates Out-of-Distribution (OOD) generalization, including unseen initial conditions and temporal extrapolation, demonstrating the operator learning architecture's superiority over standard function approximators in these scenarios.

Results

Linear Systems: On a 2D linear fractional system with 5% Gaussian noise, the proposed DeepONet-SINDy framework achieved a relative coefficient error of $0.0047 \pm 0.0039$ . This significantly outperformed the MLP ($0.0411$) and the classical FFD method, which failed completely (error $\approx 9.84$ ) due to noise amplification.
Noise Robustness: The DeepONet approach maintained high accuracy across noise levels up to 5%. As noise increased beyond this threshold, all methods degraded, but the DeepONet consistently provided the most reliable derivative estimates.
Nonlinear Systems: For nonlinear benchmarks (polynomial system and fractional Brusselator), the DeepONet reduced identification errors by over two orders of magnitude compared to FFD. Notably, even with exact derivative inputs (Oracle), nonlinear systems exhibited high errors under noise, indicating that for complex nonlinear dynamics, the limitation shifts from derivative approximation to the ill-conditioning of the sparse regression library itself.
OOD and Temporal Generalization: In tests involving initial conditions outside the training distribution and temporal extrapolation (testing on $[0, 10]$ when trained on $[0, 5]$ ), DeepONet demonstrated significantly lower error rates than the MLP, suggesting superior generalization capabilities for operator learning in this context.
Computational Efficiency: While training requires a one-time investment, the inference cost of DeepONet ($0.41$ ms per trajectory) is lower than FFD ($1.45$ ms) and scales linearly, making it efficient for repeated identification tasks within the same system family.

Significance and Claims
The paper claims that the primary obstacle in fractional system identification is not the sparse regression formulation, but the availability of reliable fractional-derivative estimates from noisy observations. By learning a data-driven surrogate for the differentiation process, the proposed framework effectively mitigates the noise sensitivity inherent in nonlocal operators.

The authors emphasize that their results provide empirical evidence that operator learning offers greater robustness to distribution shifts compared to conventional neural networks (MLPs) for this specific task. However, they modestly note that this does not constitute a proof of unique operator recovery. Furthermore, the study reveals a critical insight for nonlinear systems: once derivative estimation is sufficiently accurate, the dominant source of error becomes the sparse regression stage due to ill-conditioned libraries. Consequently, the authors suggest that future advancements for nonlinear fractional systems may depend more on robust sparse regression techniques (e.g., weak-form formulations) than on further improvements in numerical differentiation.

The work is positioned as a practical solution for scenarios where a general mathematical structure of a system family is known, allowing for offline training of the operator to be reused for multiple experimental instances without retraining. The authors acknowledge limitations, including the assumption that the fractional order $\alpha$ is known a priori and the validation being restricted to controlled synthetic data, identifying experimental validation and continuous order estimation as necessary future steps.

Technical Summary: Operator Learning for Robust Discovery of Fractional-Order Dynamical Systems from Noisy Data

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