Data-driven identification of chaotic nonlinear systems using local maximum entropy surrogates

Abstract Learning the dynamics of real-world systems without relying on explicit physical models remains a central challenge in many engineering domains. Traditional data-driven system identification frameworks either require prior knowledge about the system, such as predefined function libraries, or operate as black-box models lacking interpretability, thereby limiting their applicability in real-time systems. This paper addresses these limitations by proposing a library-free and interpretable data-driven nonlinear system identification framework based on the Local Maximum Entropy ( ) approach. The method is grounded in the principles of entropy maximization and polynomial interpolation, leveraging locally weighted basis functions with tunable priors to estimate system derivatives directly from sparse data. The study incorporates three prior functions: Gaussian, Laplacian, and Cauchy, and introduces an automated routine for selecting the optimal decay parameter, enhancing the model’s adaptability across diverse dynamical regimes. The framework is validated on simulated autonomous and non-autonomous systems, including the chaotic Lorenz system , multiple variants of the quadruple tank process (QTP) , and two real-time, sensor-instrumented physical setups: a lab-scale QTP testbed and a water distribution testbed (WDT) . Results show that the framework achieves testing MSE as low as $$\mathcal {O}(10^{-9})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>9</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for QTP configurations, and $$\mathcal {O}(10^{-4})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for highly chaotic Lorenz attractor. In real-time systems such as WDT and QTP testbeds, the framework maintains stable performance across heterogeneous sensing environments, with errors within operational tolerance. These results confirm the method’s robustness and adaptability to both synthetic and real-world dynamical systems under sparse sensing.