Effects of boundary conditions in fully convolutional networks for learning spatio-temporal dynamics

TL;DR

This paper investigates how different boundary condition strategies affect the accuracy and stability of fully convolutional networks in modeling spatio-temporal dynamics governed by PDEs, emphasizing the importance of boundary treatment.

Contribution

It systematically evaluates boundary condition strategies in FCNs for physics-based problems, highlighting the impact on network robustness and proposing improved handling methods.

Findings

Boundary implementation significantly influences accuracy and stability.

Optimal padding depends on data semantics.

Additional spatial context improves boundary handling and network robustness.

Abstract

Accurate modeling of boundary conditions is crucial in computational physics. The ever increasing use of neural networks as surrogates for physics-related problems calls for an improved understanding of boundary condition treatment, and its influence on the network accuracy. In this paper, several strategies to impose boundary conditions (namely padding, improved spatial context, and explicit encoding of physical boundaries) are investigated in the context of fully convolutional networks applied to recurrent tasks. These strategies are evaluated on two spatio-temporal evolving problems modeled by partial differential equations: the 2D propagation of acoustic waves (hyperbolic PDE) and the heat equation (parabolic PDE). Results reveal a high sensitivity of both accuracy and stability on the boundary implementation in such recurrent tasks. It is then demonstrated that the choice of the optimal padding strategy is directly linked to the data semantics. Furthermore, the inclusion of additional input spatial context or explicit physics-based rules allows a better handling of boundaries in particular for large number of recurrences, resulting in more robust and stable neural networks, while facilitating the design and versatility of such networks.