论文标题
深入学习自由边界和Stefan问题
Deep learning of free boundary and Stefan problems
论文作者
论文摘要
自由边界问题在数学，科学和工程学的许多领域都自然出现。这些问题带来了巨大的计算挑战，因为它们需要数值方法，该方法可以产生自由边界和复杂动态接口的准确近似。在这项工作中，我们提出了一个基于物理信息的神经网络的多网络模型，以解决一类称为Stefan问题的通用和反向自由边界问题。具体而言，我们近似未知的解决方案以及两个深层神经网络的任何移动边界。此外，我们制定了一种新型的反向Stefan问题，旨在直接从稀疏和嘈杂的测量中重建解决方案和自由边界。我们在一系列涵盖不同类型的Stefan问题的基准中演示了方法的有效性，并说明了建议的框架如何通过移动边界和动态接口准确地恢复部分偏微分方程的解决方案。此手稿随附的所有代码和数据均在\ url {https://github.com/predictivectiveintelligencelab/deepstefan}上公开获得。
Free boundary problems appear naturally in numerous areas of mathematics, science and engineering. These problems present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of free boundaries and complex dynamic interfaces. In this work, we propose a multi-network model based on physics-informed neural networks to tackle a general class of forward and inverse free boundary problems called Stefan problems. Specifically, we approximate the unknown solution as well as any moving boundaries by two deep neural networks. Besides, we formulate a new type of inverse Stefan problems that aim to reconstruct the solution and free boundaries directly from sparse and noisy measurements. We demonstrate the effectiveness of our approach in a series of benchmarks spanning different types of Stefan problems, and illustrate how the proposed framework can accurately recover solutions of partial differential equations with moving boundaries and dynamic interfaces. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/DeepStefan}.
