论文标题
奥登：使用人工神经网络求解普通微分方程的框架
ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks
论文作者
论文摘要
我们详细探讨了一种使用前馈神经网络求解普通微分方程的方法。我们证明了一个特定的损失函数，不需要确切解决方案的知识，它是评估神经网络性能的合适标准指标。神经网络被证明能够熟练地近似训练领域内的连续解决方案。我们说明了神经网络胜过传统标准数值技术的能力。彻底检查训练并发现了三个通用阶段：（i）先前的切线调整，（ii）曲率拟合，以及（iii）微调阶段。该方法的主要局限性是找到适当的神经网络体系结构和神经网络超参数的非平凡任务，以进行有效的优化。但是，我们观察到与微分方程的复杂性相匹配的最佳体系结构。 GitHub上提供了用户友好和适应性的开源代码（ODE $ \ MATHCAL {N} $）。
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance. Neural networks are shown to be proficient at approximating continuous solutions within their training domains. We illustrate neural networks' ability to outperform traditional standard numerical techniques. Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature fitting, and (iii) a fine-tuning stage. The main limitation of the method is the nontrivial task of finding the appropriate neural network architecture and the choice of neural network hyperparameters for efficient optimization. However, we observe an optimal architecture that matches the complexity of the differential equation. A user-friendly and adaptable open-source code (ODE$\mathcal{N}$) is provided on GitHub.
