Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders (2201.05136v1)

Abstract: A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens' theorem provides conditions for when it is possible to augment these partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original full-state system. However, the coordinate transformation back to the original attractor is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a custom deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space where it is possible to represent the dynamics in a sparse, closed form. We demonstrate this approach on the Lorenz, R\"ossler, and Lotka-Volterra systems, learning dynamics from a single measurement variable. As a challenging example, we learn a Lorenz analogue from a single scalar variable extracted from a video of a chaotic waterwheel experiment. The resulting modeling framework combines deep learning to uncover effective coordinates and the sparse identification of nonlinear dynamics (SINDy) for interpretable modeling. Thus, we show that it is possible to simultaneously learn a closed-form model and the associated coordinate system for partially observed dynamics.

• The paper presents a novel deep delay autoencoder framework that integrates time-delay embedding with SINDy to identify sparse, closed-form models of dynamic systems.
• The method accurately reconstructs underlying dynamics from limited, noisy observations, with successful validation on canonical systems like Lorenz and Rössler.
• This approach offers practical benefits for modeling incomplete data in complex fields and lays the groundwork for enhanced system identification and prediction.

Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders

The paper "Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders" provides a robust framework to address a long-standing challenge in data-driven model discovery: the reconstruction of dynamic systems from partial and often noisy observations. The authors tackle this problem by integrating deep learning techniques, namely deep autoencoders, with time-delay embedding and the Sparse Identification of Nonlinear Dynamics (SINDy) method.

Key Contributions

1. Deep Delay Autoencoders: The paper proposes a novel approach using deep autoencoders to manage high-dimensional, delay-embedded data to reveal underlying dynamic systems. The autoencoder architecture is designed to transform the delay-embedded measurement into a latent space where a simplified, interpretable model can be discovered.
2. Integration with SINDy: The integration of the deep autoencoder with the SINDy algorithm allows the authors to identify sparse, closed-form models of the dynamics. This enables the clear elucidation of governing equations that describe the system's evolution in the newly discovered coordinate system.
3. Experimental Validation: The approach is validated on several canonical examples, including the Lorenz, Rössler, and Lotka-Volterra systems, as well as a video-based dataset of a chaotic waterwheel. These examples highlight the method's ability to effectively uncover parsimonious models from partial observations.

Theoretical and Practical Implications

• Model Discovery from Partial Observations: This work has significant implications for fields that rely on incomplete data. By using a delay-embedding strategy, it allows researchers to recover essential dynamics without needing full-state observation of the system.
• Dynamical Systems Analysis: From a theoretical standpoint, the method provides an elegant solution to an open problem identified with Takens' embedding theorem; namely, determining the coordinate transformation that aligns delay embeddings with full-state representations.
• Sparse Representation: The ability to derive a sparse and interpretable model is particularly valuable for applications requiring straightforward interpretations, such as biological systems and ecological models, where complex interactions often occur.

Future Directions

The combination of autoencoders and SINDy presents numerous opportunities for further research, both in improving the scalability and efficiency of the algorithm and in expanding its applicability to more complex and noise-prone datasets. Future work could explore extensions employing richer neural network architectures, such as variational autoencoders or invertible networks, which may further improve the clarity and utility of the reconstructed coordinate systems. Additionally, enhancing the regularization techniques to ensure robust performance across a broader range of dynamical systems remains an enticing avenue.

This paper thereby sets a foundational stage for more advanced techniques in system identification and data-driven modeling, pointing towards a future where reliable models can be discerned directly from limited, real-world data, thus enabling more precise predictions and better surveillance of complex phenomena.