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

arXiv: https://arxiv.org/abs/2106.11160 · HTML: https://mlsanity.com/paper/2106.11160/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 21 Jun 2021.

## Builds on (most cited first, 10 of 10)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [An Intriguing Failing of Convolutional Neural Networks and the CoordConv Solution](https://mlsanity.com/paper/1807.03247/) (2018)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [How Much Position Information Do Convolutional Neural Networks Encode?](https://mlsanity.com/paper/2001.08248/) (2020)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Mind the Pad -- CNNs can Develop Blind Spots](https://mlsanity.com/paper/2010.02178/) (2020)
- [Partial Convolution based Padding](https://mlsanity.com/paper/1811.11718/) (2018)
- [Comparing recurrent and convolutional neural networks for predicting wave propagation](https://mlsanity.com/paper/2002.08981/) (2020)

## Built on by (most cited first, 14 of 14)

- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Performance and accuracy assessments of an incompressible fluid solver coupled with a deep Convolutional Neural Network](https://mlsanity.com/paper/2109.09363/) (2021)
- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics](https://mlsanity.com/paper/2407.03945/) (2024)
- [Predicting Wind-Driven Spatial Deposition through Simulated Color Images using Deep Autoencoders](https://mlsanity.com/paper/2202.01762/) (2022)
- [Unveiling The Mask of Position-Information Pattern Through the Mist of Image Features](https://mlsanity.com/paper/2206.01202/) (2022)
- [Predictions Based on Pixel Data: Insights from PDEs and Finite Differences](https://mlsanity.com/paper/2305.00723/) (2023)
- [Learning Physical Operators using Neural Operators](https://mlsanity.com/paper/2602.23113/) (2026)
- [Padding-free Convolution based on Preservation of Differential Characteristics of Kernels](https://mlsanity.com/paper/2309.06370/) (2023)
- [Extrapolating Phase-Field Simulations in Space and Time with Purely Convolutional Architectures](https://mlsanity.com/paper/2509.20770/) (2025)
- [Improving Translation Invariance in Convolutional Neural Networks with Peripheral Prediction Padding](https://mlsanity.com/paper/2307.07725/) (2023)
- [Physics-based machine learning for mantle convection simulations](https://mlsanity.com/paper/2505.16041/) (2025)
- [Towards Spatio-Temporal Extrapolation of Phase-Field Simulations with Convolution-Only Neural Networks](https://mlsanity.com/paper/2601.04510/) (2026)
- [Physics-informed convolutional neural networks for fluid flow through porous media](https://mlsanity.com/paper/2605.20250/) (2026)

## Abstract (arXiv)

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.