Bayesian Inference for Learning Nonlinear Dynamical Systems from Noisy and Sparse Data

מושגי ליבה

The core message of this article is to propose a novel Bayesian inference method that leverages Gaussian process modeling to learn nonlinear dynamical systems from noisy and sparse time series data, enabling robust parameter estimation and uncertainty quantification.

תקציר

The article presents a Bayesian inference method for learning nonlinear dynamical systems from time series data. The key ideas are:

Construct a vector-valued Gaussian process to represent the statistical correlation between the states and their time derivatives, preventing the need for explicit derivative evaluations.
Formulate a likelihood function that aligns the Gaussian process representation with the differential equation constraints of the dynamical system.
Perform Bayesian inference to obtain a probabilistic estimate of the system parameters, which enables uncertainty quantification in the learned model.
Discuss two scenarios: one with a given affine parametrization of the dynamics, and the other with a general nonlinear approximation using a shallow neural network.

The proposed method is demonstrated on several examples, including the Lotka-Volterra equations, a 1D nonlinear ODE, and the orbital dynamics of a black hole system. The results show that the Bayesian inference approach can provide accurate parameter estimates and reliable uncertainty quantification, especially when dealing with noisy and sparse data.

סטטיסטיקה

The Lotka-Volterra equations are given as:
˙x1 = αx1 - βx1x2
˙x2 = δx1x2 - γx2
The black hole orbital dynamics are described by the equations:
r(t) = -r(t) [cos(ϕ(t)), sin(ϕ(t))]^T
˙ϕ(t) = p/r(t)^2
˙p(t) = -e/r(t)^2

ציטוטים

"Gaussian process emulation is a probabilistic approach to supervised learning, and its variations have been applied to a wide range of computational tasks in science and engineering."
"Gaussian processes are also capable of embedding differential equation constraints into kernel structures."

תובנות מפתח מזוקקות מ:

Gaussian process learning of nonlinear dynamics

by Dongwei Ye,M... ב- arxiv.org 04-17-2024

https://arxiv.org/pdf/2312.12193.pdf

שאלות מעמיקות

How can the proposed Bayesian inference method be extended to learn dynamical systems with partial differential equation constraints

The proposed Bayesian inference method can be extended to learn dynamical systems with partial differential equation (PDE) constraints by incorporating the PDE constraints into the likelihood function. When dealing with systems governed by PDEs, the state variables and their derivatives can be represented as Gaussian processes, similar to the approach used for ordinary differential equations. The correlation between the states and their derivatives can be modeled using Gaussian processes, allowing for the inclusion of PDE constraints in the inference process.
To extend the method to PDE-constrained systems, the likelihood function can be formulated to capture the alignment between the observed data and the solutions of the PDEs. The parameters of the PDEs can be inferred through Bayesian inference, considering the uncertainties in the data and the model. By incorporating the PDE constraints into the Bayesian framework, the method can effectively learn the dynamics of systems governed by PDEs, providing a probabilistic estimate of the model parameters while quantifying uncertainties.

What are the potential limitations of the neural network parametrization approach, and how can it be improved further

The neural network parametrization approach, while powerful in capturing complex nonlinear relationships, has certain limitations that can affect its performance in certain scenarios. Some potential limitations of the neural network approach include:

Limited Generalization: Neural networks may struggle to generalize well outside the range of the training data, leading to poor performance in extrapolation tasks.

Overfitting: Neural networks are prone to overfitting, especially when the model complexity is high relative to the amount of training data available. This can result in poor performance on unseen data.

Interpretability: Neural networks are often considered as black-box models, making it challenging to interpret how the model arrives at its predictions. This lack of interpretability can be a drawback in certain applications where explainability is crucial.

To improve the neural network parametrization approach further, techniques such as regularization, dropout, early stopping, and model architecture optimization can be employed to mitigate overfitting and enhance generalization. Additionally, incorporating domain knowledge into the neural network design and utilizing techniques like attention mechanisms can help improve interpretability and performance.

Can the Bayesian inference framework be combined with other data-driven techniques, such as sparse identification of nonlinear dynamics, to enhance the interpretability and generalization of the learned models

The Bayesian inference framework can be combined with sparse identification of nonlinear dynamics (SINDy) to enhance the interpretability and generalization of the learned models. By integrating Bayesian inference with SINDy, the following benefits can be achieved:

Interpretability: Bayesian inference provides a probabilistic framework for parameter estimation, allowing for uncertainty quantification in the model predictions. When combined with SINDy, which identifies sparse representations of the dynamics, the resulting model becomes more interpretable as it captures the essential dynamics with a reduced set of terms.

Generalization: The Bayesian framework can help in generalizing the learned models by incorporating uncertainties in the data and the model parameters. This leads to more robust models that can make reliable predictions even in the presence of noise or limited data.

Model Validation: The combination of Bayesian inference and SINDy enables rigorous model validation by quantifying uncertainties and assessing the reliability of the learned dynamical models. This validation process enhances the trustworthiness of the models and their predictions.

By integrating Bayesian inference with sparse identification techniques like SINDy, the resulting models benefit from both the interpretability of sparse representations and the robustness of Bayesian uncertainty quantification, leading to more reliable and generalizable data-driven models of dynamical systems.