Learning the temporal evolution of multivariate densities via normalizing flows

Area of Science:

• Computational Mathematics and Statistics
• Machine Learning for Scientific Computing
• Stochastic Processes and Differential Equations

Background:

• Understanding temporally evolving probability distributions is crucial in various scientific fields.
• Traditional methods struggle with high-dimensional, complex distributions generated by stochastic differential equations (SDEs).
• Fokker-Planck equations describe the evolution of probability densities but can be computationally intensive to solve directly.

Purpose of the Study:

• To develop a novel machine learning-based method for learning multivariate probability distributions from SDE sample path data.
• To construct a time-dependent mapping that transforms a reference distribution into evolving density snapshots.
• To approximate probability density function (PDF) evolution over time for systems driven by Brownian and Lévy noise.

Main Methods:

• Utilized sample path data generated by stochastic differential equations.
• Employed machine learning to construct a time-dependent mapping, specifically a multivariate normalizing flow.
• The normalizing flow deforms a reference distribution (e.g., Gaussian) to match target density snapshots at different times.

Main Results:

• Successfully learned and approximated time-dependent probability distributions from SDE data.
• Demonstrated the method's ability to capture PDF evolution for systems with both Brownian and Lévy noise.
• Validated the approach using two- and three-dimensional examples, including uni- and multimodal distributions.

Conclusions:

• The proposed normalizing flow method provides an effective way to learn complex, evolving probability distributions from SDEs.
• This approach offers a powerful tool for analyzing and simulating dynamic systems described by stochastic processes.
• The method shows promise for applications requiring accurate modeling of time-varying probability densities.