Learning dynamics on invariant measures using PDE-constrained optimization.

Area of Science:

• Dynamical Systems Theory
• Scientific Machine Learning
• Computational Physics

Background:

• Learning autonomous continuous-time dynamical systems is crucial for modeling complex phenomena.
• Existing methods often require densely sampled data, limiting their applicability.
• Invariant measures provide a powerful statistical description of system dynamics.

Purpose of the Study:

• To extend existing methodologies for learning dynamical systems from invariant measures.
• To reformulate the inverse problem of learning Ordinary Differential Equations (ODEs) or Stochastic Differential Equations (SDEs) as a PDE-constrained optimization problem.
• To enable learning from slowly sampled data and perform uncertainty quantification.

Main Methods:

• Reformulation of the inverse problem as a PDE-constrained optimization.
• Utilizing invariant measures for system identification.
• Developing a forward model with enhanced stability properties.

Main Results:

• Successfully learned autonomous continuous-time dynamical systems from invariant measures.
• Demonstrated effective learning from slowly sampled inference trajectories.
• Achieved uncertainty quantification for forecasted dynamics.
• Showcased improved forward model stability compared to direct simulation in specific cases.

Conclusions:

• The proposed PDE-constrained optimization approach offers a robust and versatile framework for learning dynamical systems.
• The method is effective for both benchmark systems (Van der Pol, Lorenz-63) and real-world applications (Hall-effect thruster, temperature prediction).
• This approach enhances the ability to model and predict complex dynamics from limited observational data.