Bayesian reasoning for physics-informed neural networks.

Area of Science:

• Computational Physics
• Machine Learning
• Numerical Analysis

Background:

• Physics-informed neural networks (PINNs) integrate physical laws into neural network training.
• Current Bayesian PINN methods often rely on computationally expensive sampling or variational inference.
• Optimizing the balance between data fidelity and physical constraints in PINNs remains a challenge.

Purpose of the Study:

• To develop an evidence-driven Bayesian formulation of PINNs for automatic loss weight optimization.
• To enable efficient hyperparameter tuning and model comparison without posterior sampling.
• To provide a unified Bayesian framework for integrating governing equations and observational data.

Main Methods:

• An evidence-driven Bayesian formulation of PINNs is introduced.
• Laplace approximation is employed for analytical computation of model evidence.
• The method is demonstrated on the heat, wave, and Burgers' equations.

Main Results:

• The proposed method achieves automatic optimization of loss weights for PDE residuals, boundary conditions, and observational data.
• Solutions obtained for benchmark PDEs (heat, wave, Burgers') agree with exact or reference results.
• The framework successfully integrates governing equations and noisy measurements, yielding predictive uncertainties.

Conclusions:

• The novel Bayesian PINN formulation offers an efficient and robust approach for solving differential equations.
• Analytical computation of model evidence facilitates streamlined hyperparameter tuning and model selection.
• The method's ability to provide predictive uncertainties enhances its applicability in real-world scientific problems.