# Guiding continuous operator learning through Physics-based boundary constraints

**Authors:** Nadim Saad, Gaurav Gupta, Shima Alizadeh, Danielle C. Maddix  
**arXiv:** [2212.07477](https://arxiv.org/abs/2212.07477)  
**Date:** 2022-12-14  
**Summary page:** https://www.summarizepaper.com/en/paper/2212.07477v2/

## Key points

- Partial Differential Equations (PDEs) are crucial for modeling physical phenomena in scientific and engineering applications.
- Boundary Conditions (BCs) are constraints imposed at specific spatial locations to ensure solution validity in PDEs.
- Traditional approaches enforce BCs explicitly during solution procedures, while neural-network based methods learn BCs implicitly through training data.
- The novel approach called Boundary enforcing Operator Network (BOON) modifies operator kernels to ensure neural operators satisfy physics-based BCs like Dirichlet, Neumann, and periodic conditions.
- BOON demonstrates enhanced accuracy and ensures BC satisfaction across various PDE applications, leading to significant improvements in solution accuracy over the entire domain compared to existing operator models.

## Summary

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In the realm of scientific and engineering applications, Partial Differential Equations (PDEs) play a crucial role in modeling various physical phenomena. These PDEs often come with Boundary Conditions (BCs), which are constraints imposed at specific spatial locations to ensure the validity of solutions. For instance, in the heat equation describing heat transfer, BCs dictate the behavior of temperature gradients at boundaries, such as insulators or heaters. Similarly, boundary conditions like no-slip for viscous flows and periodic for turbulent flows are essential in computational fluid dynamics. The adherence to these BCs is paramount as violating them can lead to unstable models and non-physical results. Traditional approaches rely on explicit enforcement of BCs during solution procedures. However, recent advancements in neural-network based methods have introduced implicit learning of BCs through training data. While effective, these models lack a guarantee of BC satisfaction during evaluation. To address this limitation, a novel approach called Boundary enforcing Operator Network (BOON) has been proposed. BOON modifies the structure of operator kernels to ensure neural operators satisfy physics-based BCs like Dirichlet, Neumann, and periodic conditions. Through a refined procedure, BOON demonstrates enhanced accuracy and ensures BC satisfaction across various PDE applications. Numerical experiments showcasing the effectiveness of BOON reveal significant improvements in solution accuracy over the entire domain compared to existing operator models. Notably, the proposed correction method exhibits substantial enhancements ranging from 2X to 20X reduction in relative $L^2$ error for equations like Burgers' equation. This innovative framework presented by Nadim Saad et al., published at ICLR 2023, marks a significant advancement in guiding continuous operator learning through physics-based boundary constraints. By integrating structural changes into neural operators to enforce BC satisfaction explicitly, BOON offers a promising avenue for achieving more accurate solutions while upholding critical physics-enforced constraints in PDE modeling.