LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systems

Abstract

Sparse Identification of Nonlinear Dynamical Systems (SINDy) is a powerful tool for the data-driven discovery of governing equations. However, it encounters challenges when modeling complex dynamical systems involving high-order derivatives or discontinuities, particularly in the presence of noise. These limitations restrict its applicability across various fields in applied mathematics and physics. To mitigate these, we propose Laplace-Enhanced SparSe Identification of Nonlinear Dynamical Systems (LES-SINDy). By transforming time-series measurements from the time domain to the Laplace domain using the Laplace transform and integration by parts, LES-SINDy enables more accurate approximations of derivatives and discontinuous terms. It also effectively handles unbounded growth functions and accumulated numerical errors in the Laplace domain, thereby overcoming challenges in the identification process. The model evaluation process selects the most accurate and parsimonious dynamical systems from multiple candidates. Experimental results across diverse ordinary and partial differential equations show that LES-SINDy achieves superior robustness, accuracy, and parsimony compared to existing methods.

Figures (5)

• Figure 1: Illustration of the LES-SINDy framework, demonstrated on identifying the Lorenz system. The process starts by extracting time-series measurements from sensors, which are used to form a Laplace-enhanced library. Sparse regression techniques are then employed to identify potential models in the Laplace domain. The evaluation stage determines the most accurate and parsimonious model, which successfully recovers the Lorenz system.
• Figure 2: Simulation of the High-Order Linear ODE Systems and the corresponding log RMSE mappings. Top Left: Dynamic behavior of the Duffing oscillator across various damping ratios $\zeta$. The x-axis denotes the time, and the y-axis represents the magnitude of the dynamical system responses. When $\zeta=0.20$, it represents a lightly damped system, where oscillations will persist with gradual decay. $\zeta=0.50$ represents a moderately damped system, where the oscillations decay faster than in the lightly damped case. For $\zeta=1.0$, the system returns to equilibrium as quickly as possible without oscillating. Overdamped responses are demonstrated at $\zeta = 2.0$ and $\zeta = 6.0$, with increasingly slow returns to equilibrium, which indicates higher damping levels. Top Right: Dynamic behavior of a fourth-order linear ODE. The response implies quasi-periodic motion, which is characterized by a combination of oscillations and a time-dependent modulation of amplitude. This results in oscillations whose amplitude gradually increases over time, which reflects a deterministic and predictable pattern. The system exhibits intricate behavior where the oscillations shift in both amplitude and phase and demonstrates the dynamic interplay between periodicity and linear modulation as time progresses. Bottom Left: log RMSE Error Mapping for the Duffing Oscillators: Noise Scale and $\zeta$ Value Effects in Logarithmic Scale. The x-axis denotes the selection of the noise scale, the y-axis is the magnitude of $\zeta$, and the colors represent the log RMSE errors between the predicted results and the ground truth. Magnitudes of noise scale increase uniformly from 0.0 to 0.20, and the $\zeta$ increases exponentially from $e^{-2}\approx 0.135$ to $e^2\approx 7.389$. The heat mapping indicates the $\zeta$ value may affect the result of good predictions. Bottom Right: log RMSE Error Mapping for a Homogeneous Linear Fourth-Order ODEs: Initial Value and Step Size Effects in Logarithmic Scale. The x-axis denotes the selection of the initial value $s_1$ (intercept), the y-axis is the interval between different $s$ values (step size), and the colors represent the log RMSE errors between the predicted results and the ground truth. Values of $s$ are uniformly distributed from $s_1$ to $s_{20}$ based on the chosen intercept and the step size. The heat mapping indicates the appropriate selection of intercept and step size is crucial for good predictions.
• Figure 3: Top Left: Simulation of the ODEs with Discontinuous Inputs. The x-axis denotes the time, and the y-axis represents the magnitude of the dynamical system response. Top Right: Simulation of the ODEs with Trigonometric (sine and cosine) and Hyperbolic (sinh and cosh) Functions. Bottom: log RMSE Error Mapping: Noise Scale and # Measurements Effects (Tested in ODEs with Trigonometric Inputs) in Logarithmic Scale. The x-axis denotes the noise scale of the measurements, the y-axis represents the number of measurements used for LES-SINDy, and the colors represent the log RMSE errors between the predicted results and the ground truth. Magnitudes of noise scale increase uniformly from $0.0$ to $0.11$, and the number of measurements increases exponentially from $2^3$ to $2^{13}$. The color depth in heat mapping indicates the change in prediction accuracy as the amount of measurement noise and the amount of measurement increase. The left figure denotes the log RMSE error mapping for an ODE with a sine input, and the right one represents the results of an ODE with a cosine input.
• Figure 4: Simulation of Nonlinear ODE systems. Left: Simulation of the Lorenz system's state variables $u=\left[x, y, z\right]^\intercal$. The x-axis denotes the time, and the y-axis represents the magnitude of the dynamical system. Middle: The corresponding trajectory of the Lorenz system in the three-dimensional phase space. Right: Simulation of the Lotka–Volterra model. The x-axis denotes the time, and the y-axis represents the population dynamics of prey and predator.
• Figure 5: Identified models compared with the ground truth for canonical PDEs, tested with different initial conditions. Top Left: the convection-diffusion equation. Top Right: the Burgers equation. Bottom: Kuramoto-Sivashinsky equation.