Robust Symbolic Regression for Identifying Nonlinear Dynamics from Sparse and Noisy Data using Gaussian Processes

Основные понятия

A method combining Gaussian process regression and sparse identification of nonlinear dynamics (SINDy) to learn accurate analytical models of nonlinear dynamical systems from sparse and noisy data.

Аннотация

The paper addresses the challenge of deriving dynamical models from sparse and noisy data. It proposes a method called GPSINDy that combines Gaussian process regression and the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm to denoise the data and identify nonlinear dynamical equations.

Key highlights:

• Gaussian process regression is used to smooth and interpolate the sparse, noisy state measurements, providing better estimates of the state derivatives for symbolic regression.
• The smoothed state and derivative estimates are then used in the LASSO optimization problem of SINDy to discover the governing equations.
• The method is benchmarked on a Lotka-Volterra model, a unicycle dynamic model in simulation, and hardware data from an NVIDIA JetRacer system.
• Experiments show GPSINDy outperforms baselines like standard SINDy and neural network-based methods, achieving up to 61.92% improvement in predicting future trajectories from the discovered dynamics.
• The novelty lies in the use of Gaussian processes to smooth the state measurements and compute the state derivatives, which improves the robustness of symbolic regression to noise and data sparsity.

Статистика

The Lotka-Volterra model has the following true coefficients: a = 1.1, b = 0.4, c = 1.0, d = 0.4.
For the unicycle model, the control inputs are u1(t) = sin(t) and u2(t) = 0.5 cos(t).
The NVIDIA JetRacer data was collected at 50 Hz for 22.85 seconds, with the state consisting of x-y position, forward velocity, and heading angle.

Цитаты

"Our simple approach offers improved robustness with sparse, noisy data compared to SINDy alone."
"Gaussian process regression is particularly effective as an interpolation tool and at reducing the noise in measurement data."
"GPSINDy consistently outperforms the baselines across all noise magnitudes, highlighting its robustness in dealing with noisy data."

Ключевые выводы из

Symbolic Regression on Sparse and Noisy Data with Gaussian Processes

by Junette Hsin... в arxiv.org 03-29-2024

https://arxiv.org/pdf/2309.11076.pdf

Дополнительные вопросы

How can the proposed GPSINDy method be extended to handle more complex nonlinear dynamics, such as those with discontinuities or time-varying parameters

To extend the GPSINDy method to handle more complex nonlinear dynamics with discontinuities or time-varying parameters, several modifications and enhancements can be implemented. One approach is to incorporate adaptive kernel functions in the Gaussian process regression to better capture discontinuities in the data. By using kernels that are specifically designed to handle abrupt changes or non-smooth behavior in the data, the model can adapt to these complexities more effectively. Additionally, introducing time-varying parameters into the Gaussian process regression can allow the model to capture dynamic changes in the system over time. This can be achieved by incorporating time-dependent hyperparameters in the kernel functions, enabling the model to adapt to evolving dynamics. By combining these adaptive techniques with the GPSINDy framework, the method can be extended to handle a wider range of nonlinear dynamics with discontinuities and time-varying parameters.

What are the limitations of using Gaussian processes for data smoothing, and how can they be addressed to further improve the performance of symbolic regression on real-world systems

While Gaussian processes are effective for data smoothing and interpolation, they have certain limitations that can impact their performance in symbolic regression on real-world systems. One limitation is the computational complexity associated with Gaussian process regression, especially as the size of the dataset increases. This can lead to scalability issues when dealing with large datasets or high-dimensional data. To address this limitation, techniques such as sparse Gaussian processes or approximations like variational inference can be employed to reduce the computational burden while maintaining accuracy. Another limitation is the sensitivity of Gaussian processes to the choice of kernel functions. In cases where the data exhibits complex patterns or structures that are not well captured by standard kernels, custom kernel functions tailored to the specific characteristics of the data can be developed. By addressing these limitations and optimizing the Gaussian process regression for the specific requirements of symbolic regression on real-world systems, the performance of the GPSINDy framework can be further improved.

Can the GPSINDy framework be integrated with other techniques, such as physics-informed neural networks, to leverage both data-driven and physics-based modeling approaches

The GPSINDy framework can be integrated with other techniques, such as physics-informed neural networks (PINNs), to leverage both data-driven and physics-based modeling approaches effectively. By combining the strengths of Gaussian process regression for data smoothing and interpolation with the flexibility and scalability of neural networks, the integrated framework can offer a comprehensive solution for system identification and modeling. In this integrated approach, the Gaussian process regression can be used to preprocess the data and provide smoothed inputs for the physics-informed neural network. The PINN can then incorporate the learned dynamics from GPSINDy as constraints or regularization terms in the neural network training process, ensuring that the model adheres to the underlying physics of the system. This integration allows for the incorporation of domain knowledge and physical constraints into the data-driven modeling process, enhancing the interpretability and accuracy of the learned models. By combining GPSINDy with physics-informed neural networks, the framework can achieve a more robust and comprehensive approach to symbolic regression on real-world systems.