Particles to partial differential equations parsimoniously

Area of Science:

• Computational Physics
• Data Science
• Applied Mathematics

Background:

• Physico-chemical processes are typically described by microscopic equations.
• Effective equations at macroscopic scales (meso- or macroscopic) can simplify complex systems.
• Discovering these coarse-grained equations aids in computational prediction and control.

Purpose of the Study:

• To develop an efficient framework for discovering macro-scale partial differential equations (PDEs) from microscopic simulations.
• To reduce the computational cost associated with data collection for training machine learning models.
• To identify suitable macro-scale variables for formulating coarse-grained effective PDEs.

Main Methods:

• Combining artificial neural networks with equation-free numerics for multiscale computation.
• Employing manifold learning techniques, including unnormalized optimal transport and moment-based distribution descriptions.
• Utilizing sparse data collection in the space-time domain.

Main Results:

• Efficient discovery of coarse-grained PDEs directly from microscopic simulations.
• Significant reduction in the computational effort required for data collection.
• Successful identification of macro-scale variables, corroborating or introducing new candidates.

Conclusions:

• The proposed framework effectively extracts coarse-grained evolution equations from particle-based simulations.
• The integration of equation-free numerics and data-driven manifold learning offers a computationally parsimonious approach.
• This method facilitates the discovery of effective macro-scale PDEs for systems with unknown macroscopic variables.