Physics-Informed Neural Networks with Dynamical Boundary Constraints

TL;DR

This paper introduces the Dynamical Boundary Constraint (DBC) algorithm to improve physics-informed neural networks (PINNs) by addressing convergence issues and enhancing their ability to handle multi-scale and oscillatory problems.

Contribution

The paper proposes a novel DBC algorithm that incorporates prior training to improve PINN convergence and accuracy in complex physical systems.

Findings

DBC improves PINN convergence on multi-scale problems

Enhanced accuracy in oscillatory differential equations

Demonstrated applicability across various physics domains

Abstract

Physics-informed neural networks (PINNs) are numerical solvers that embed all the physical information of a system into the loss function of a neural network. In this way the learned solution accounts for data (if available), the governing differential equations, or any other constraint known of the physical problem. However, they face serious issues, notably their tendency to converge on trivial or misleading solutions. The latter occurs when, although the loss function reaches low values the model makes incorrect predictions. These difficulties become especially significant in differential equations involving multi-scale behavior, such as rapidly varying terms and solutions exhibiting strong oscillatory behavior. To address these challenges, we introduce the Dynamical Boundary Constraint (DBC) algorithm, which imposes restrictions on the loss function based on prior training of the PINN. To demonstrate its applicability, we tested this approach on examples of different areas of physics.