用可满足边界条件的神经网络结构，避免传统方法依赖超参惩罚项。
A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems
- 设计新神经网络结构，自动满足变分问题边界条件
- 无需惩罚项，优化过程更稳定，误差更低
- 理论基于Sobolev空间，为方法提供严格数学支撑
近期，结合深度学习的Ritz方法（即Deep Ritz Method）被提出，利用神经网络作为变分问题的试函数。然而，神经网络本身不天然满足边界条件，该方法通常引入依赖超参数的惩罚项，可能导致优化过程出现误导性结果。本文提出一种新试函数（ansatz），其构造天然满足变分问题的边界条件，从而完全消除对惩罚项的需求。研究的关键贡献在于：所有支撑定理与推论均在Sobolev范数下建立，这构成了变分问题的自然框架，因泛函显式依赖解及其导数。该理论保证了所提ansatz在Ritz方法中的表达能力与适定性。实验表明，该方法不仅避免了误导性优化结果，还降低了复杂度并保持高精度，体现出解决变分问题的实用有效性。
原文摘要 · Abstract (English)
Recently, innovative adaptations of the Ritz method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the trial function for variational problems. However, the neural network does not inherently satisfy the boundary conditions of the variational problem. To address this issue, the Deep Ritz Method introduces a penalty term into the functional, which is strongly dependent on hyperparameters and may lead to misleading results during the optimization process. In this work, we propose an ansatz that inherently satisfies the boundary conditions of the variational problem, thereby eliminating the need for penalty terms. A key contribution of this study is that all supporting theorems and corollaries are established in Sobolev norms, which constitute the natural framework for variational problems, as the functional depends explicitly on the solution and its derivatives. This provides a rigorous justification for the expressiveness and admissibility of the proposed ansatz within the Ritz method. The results demonstrate that the proposed ansatz not only avoids misleading optimization outcomes but also reduces complexity while maintaining accuracy, highlighting its practical effectiveness for solving variational problems.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。