Peter Y. Lu | Discovering Sparse Interpretable Dynamics

Discovering sparse interpretable dynamics from partial observations

Peter Y. Lu, Joan Ariño Bernad, and Marin Soljačić

Communications Physics (2022)

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Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data. Achieving this kind of interpretable system identification is even more difficult for partially observed systems. We propose a machine learning framework for discovering the governing equations of a dynamical system using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model. The entire architecture is trained end-to-end by matching the higher-order symbolic time derivatives of the sparse symbolic model with finite difference estimates from the data. Our tests show that this method can successfully reconstruct the full system state and identify the equations of motion governing the underlying dynamics for a variety of ordinary differential equation (ODE) and partial differential equation (PDE) systems.

Complex dynamical systems, such as weather, climate, chemical, and biological systems, are often hard to analyze, understand, and model. Identifying the governing equations of a nonlinear dynamical system is key to both understanding the physical features of the system and constructing an accurate model of the dynamics that generalizes well beyond the available data.

However, in many real-world scenarios, there are some quantities that are easy to measure, such as the concentration of a fluorescent dye, and others that are very difficult, expensive, or simply impossible to observe directly. That is, we are often only able to partially observe the state of system, providing us with incomplete data and further complicating the modeling process.

Our machine learning approach addresses these issues by reconstructing the original system state from partial observations while, at the same time, fitting an interpretable symbolic equation to model the system dynamics.

Discovering sparse interpretable dynamics from partial observations.

Architecture

We propose a machine learning framework for discovering the governing equations of a dynamical system using only partial observations, combining an encoder for state reconstruction with a sparse symbolic model.

Machine learning framework for simultaneously reconstructing hidden states and fitting a symbolic model.

Experiments

Our tests show that our method can successfully reconstruct the full system state and identify the underlying dynamics for a variety of ODE and PDE systems. We test our method on partially observed chaotic dynamics in the form of the Rössler and Lorenz systems.

System identification and hidden state reconstruction for ODE systems.

Our method is able to reconstruct the full dynamics of a 2D diffusive spatiotemporal system with a hidden dynamic source, as well as a partially observed 2D reaction–diffusion system.

System identification and hidden state reconstruction for PDE systems.

As an additional example, our method is also able to handle missing phase information $\varphi = \arg \psi$ and identify the dynamics of a complex nonlinear wave equation.

System identification and phase reconstruction for nonlinear wave propagation.

Prediction Examples

As a result of reconstructing the hidden states and identifying the exact symbolic dynamics, our fitted models are able predict and generalize extremely well.

Prediction examples on test data from the Lorenz system.

\tau

is the Lyapunov time.

Prediction examples on test data from the diffusive Lokta–Volterra system showing (a) snapshots of the visible state

u

and (b) the spatial average of

u

over time.

This new framework shows how machine learning can generate interpretable models even from incomplete measurements and will enable us to better understand real-world data from complex dynamical systems.

Please see our paper for more details.