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This study introduces methods to automatically discover governing equations from data, integrating them into Physics-Informed Neural Networks (PINNs) and Bayesian approaches. This enhances predictive accuracy for complex systems, even with incomplete information.

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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.