Robust Discovery of Open-form Partial Differential Equations from Limited and Noisy Data

Kernekoncepter

A robust framework that seamlessly integrates symbolic mathematics and automatic equation embedding to discover open-form partial differential equations from limited and noisy data.

Resumé

The proposed R-DISCOVER framework consists of two primary procedures: discovering and embedding.

Discovering Process:

1. The framework utilizes a RL-guided hybrid PDE generator to efficiently generate diverse open-form PDE expressions represented as binary tree structures.
2. A neural network-based predictive model is built to fit the system response and serve as a reward evaluator for the generated PDE expressions.
3. The RL agent is updated using the risk-seeking policy gradient method with the better-fitting PDE expressions.
4. A parameter-free model selection method is proposed to determine the initially identified PDE by balancing data fitness and coefficient stability.

Embedding Process:

1. The initially identified PDE is automatically incorporated as a physical constraint into the neural network-based predictive model by traversing the PDE tree.
2. The predictive model is then optimized with the discovered PDE constraint and the observed data, enhancing the robustness to noise.

The alternating updates of the discovering and embedding processes enable the framework to uncover accurate governing equations from nonlinear dynamic systems with limited and highly noisy data, outperforming other physics-informed neural network-based discovery methods.

Statistik

The relative error of the coefficients between the uncovered and true equations can be as low as 5%.
The true positive rate, which measures the accuracy of the identified equation form, can reach 1, indicating the correct equation form has been successfully retrieved.

Citater

"The proposed framework seamlessly integrates symbolic mathematics and automatic equation embedding to robustly discover open-form partial differential equations from nonlinear systems."
"The RL-guided hybrid PDE generator can efficiently generate diverse open-form PDE expressions, while the automatic embedding of discovered equations enhances the robustness to noise."
"The alternating updates of discovering and embedding processes enable the framework to uncover accurate governing equations from limited and highly noisy data."

Vigtigste indsigter udtrukket fra

Physics-constrained robust learning of open-form partial differential equations from limited and noisy data

by Mengge Du,Yu... kl. arxiv.org 04-30-2024

https://arxiv.org/pdf/2309.07672.pdf

Dybere Forespørgsler

How can the proposed framework be extended to handle systems with multiple coupled PDEs

To extend the proposed framework to handle systems with multiple coupled PDEs, we can modify the symbolic representation and the PDE discovery process. One approach is to introduce a hierarchical structure in the symbolic representation to account for the interconnections between multiple PDEs. Each PDE can be represented as a subtree within a larger tree structure, reflecting the coupling between different equations. The PDE generator can then be adapted to generate and optimize these interconnected PDE expressions simultaneously. By incorporating cross-references and dependencies between the PDEs in the reward evaluation process, the framework can identify the governing equations of complex systems with multiple coupled PDEs.

What are the potential limitations of the symbolic representation and how can they be addressed

One potential limitation of symbolic representation is the scalability and complexity of the generated PDE expressions. As the number of variables and terms in the equations increases, the search space grows exponentially, making it challenging to efficiently explore all possible combinations. To address this limitation, techniques such as hierarchical representation, constraint propagation, and domain-specific knowledge can be utilized to reduce the search space and guide the generation of meaningful PDE expressions. Additionally, incorporating domain-specific constraints and regularization techniques can help prevent the generation of overly complex or redundant equations, improving the efficiency and effectiveness of the framework.

Can the framework be applied to discover governing equations in real-world applications with limited prior knowledge and understanding

The proposed framework can be applied to discover governing equations in real-world applications with limited prior knowledge and understanding by leveraging the flexibility and adaptability of symbolic representation and reinforcement learning. By allowing the system to explore a wide range of equation forms and structures, the framework can uncover hidden relationships and patterns in the data, even in the absence of comprehensive domain knowledge. Additionally, the automatic physics embedding process ensures that the discovered equations are integrated into the predictive model, enhancing the robustness and accuracy of the predictions. By iteratively updating the discovery and embedding processes, the framework can adapt to diverse and complex systems, making it suitable for real-world applications where limited understanding is a common challenge.