Physics-informed neural ODE (PINODE): embedding physics into models using collocation points.

Area of Science:

• Dynamical Systems and Control
• Scientific Machine Learning
• Numerical Analysis

Background:

• Reduced-order models (ROMs) are crucial for analyzing complex dynamical systems.
• Autoencoder-based ROMs require substantial data, limiting their application in data-scarce environments.
• Integrating physics knowledge into data-driven models is an active research area.

Purpose of the Study:

• To develop a data-efficient method for building reduced-order models (ROMs) by incorporating physics knowledge.
• To improve the performance of ROMs in low-data regimes and noisy conditions.
• To enhance the utility of latent-space dimensions and out-of-distribution forecasting capabilities.

Main Methods:

• A collocation-based physics-informed loss term is proposed to embed known physical equations into the latent-space dynamics of ROMs.
• The method leverages classical collocation techniques from numerical analysis.
• The approach is tested on a high-dimensional nonlinear partial differential equation (PDE).

Main Results:

• Significant performance gains were observed in low-data regimes (up to 5x improvement in prediction error).
• Substantial improvements were achieved in high-noise learning (up to 10x) and latent-space dimension utilization (up to 100x).
• Exceptional gains (up to 200x) were recorded in far-out out-of-distribution forecasting compared to purely data-driven models.

Conclusions:

• The proposed physics-informed loss term enables effective ROM training even with limited data.
• This approach enhances the robustness and predictive power of ROMs across various challenging scenarios.
• The findings facilitate broader application of physics-informed ROMs in fields like compressive sensing and control.