Physics-Informed Neural Networks with Unknown Partial Differential Equations: An Application in Multivariate Time...

Area of Science:

• Artificial Intelligence
• Computational Science
• Applied Mathematics

Background:

• Neural Networks (NNs) can integrate domain knowledge via custom loss functions to improve predictions with limited data.
• Physics-Informed Neural Networks (PINNs) use Partial Differential Equations (PDEs) as constraints to guide NNs.
• Bayesian Neural Networks (BNNs) extend this for uncertainty quantification, but require known governing equations.

Purpose of the Study:

• To develop methods for automatically selecting PDEs from historical data when governing equations are unknown.
• To integrate these discovered PDEs into advanced modeling frameworks: PINNs, Bayesian-PINNs (B-PINNs), and Physical-Informed Bayesian Linear Regression (PI-BLR).
• To evaluate the effectiveness of these physics-guided machine learning approaches on a real-world energy management dataset.

Main Methods:

• Automated selection of parametric PDEs from historical Multivariate Time Series (MTS) data.
• Integration of discovered PDEs into PINN, B-PINN, and PI-BLR frameworks.
• Comparative evaluation of model performance in forecasting future states under varying data conditions and constraint scenarios.

Main Results:

• Demonstrated ability to automatically learn governing equations from data.
• Successful integration of learned PDEs into multiple physics-informed modeling approaches.
• Comparative analysis showing the impact of PDE constraints on forecasting accuracy in an energy management context.

Conclusions:

• Physics-guided machine learning frameworks can be enhanced by automatically discovered equations.
• These methods offer a pathway to improve predictions in systems with unknown or partially known dynamics.
• The research bridges data-driven discovery and physics-based modeling for practical applications.