Physics-informed neural networks for solving derivative-constrained partial differential equations

Area of Science:

• Computational Science
• Applied Mathematics
• Machine Learning

Background:

• Physics-informed neural networks (PINNs) solve PDEs via optimization but often miss fundamental derivative-based constraints.
• Many real-world applications necessitate incorporating complex relationships involving states and their derivatives.

Purpose of the Study:

• Introduce Derivative-Constrained PINNs (DC-PINNs) as a general framework for solving constrained PDEs.
• Embed nonlinear derivative constraints efficiently using automatic differentiation.
• Develop self-adaptive loss balancing to reduce hyperparameter tuning.

Main Methods:

• Formulate constrained PDE solving as an optimization problem guided by a minimum objective function.
• Incorporate general nonlinear constraints on states and derivatives (e.g., bounds, monotonicity, incompressibility).
• Utilize automatic differentiation for efficient constraint computation and self-adaptive loss balancing for objective weighting.

Main Results:

• DC-PINNs demonstrate consistent reduction in constraint violations and improved physical fidelity compared to baseline PINNs.
• Successfully applied to benchmarks including heat diffusion with bounds, arbitrage-free financial volatilities, and vortex shedding in fluid flow.
• Showcase improved training stability and convergence towards physically admissible solutions, even with small PDE residuals.

Conclusions:

• DC-PINNs provide a robust and generalizable approach for solving constrained PDEs.
• Explicitly encoding derivative constraints enhances the reliability and physical interpretability of solutions.
• The framework offers a path towards more accurate and stable physics-informed machine learning models grounded in physical principles.