2 posts tagged with "spectral methods"

这是一篇开山之作提出了一个深度学习框架 DeepONet 用于求解偏微分方程的求解器，这篇论文介绍了原理。

摘要​

摘要原文：

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator [5]. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors $x_i = 1, ... , m$ (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.

摘要翻译：

尽管神经网络是连续函数的通用逼近器这一事实广为人知，但一个较少为人所知且可能更强大的结果是：具有单个隐藏层的神经网络能够精确逼近任何非线性连续算子。这一通用逼近定理暗示了神经网络在从数据中学习非线性算子方面的潜在应用。然而，该定理仅保证在网络规模足够大时存在较小的逼近误差，并未考虑重要的优化误差和泛化误差。为了在实践中实现这一定理，我们提出**深度算子网络（DeepONets）**以从相对较小的数据集准确高效地学习算子。一个 DeepONet 由两个子网络组成：一个用于在固定数量的传感器上编码输入函数 $x_i = 1, ..., m$（分支网络），另一个用于编码输出函数的位置（主干网络）。我们通过系统性模拟识别两种类型的算子，即动态系统和偏微分方程，并证明 DeepONet 相较于全连接网络显著降低了泛化误差。我们还从理论上推导了近似误差与传感器数量（即输入函数定义的传感器数量）以及输入函数类型之间的依赖关系，并通过计算结果验证了该定理。更重要的是，我们在计算测试中观察到高阶误差收敛，即多项式收敛率（从半阶到四阶）甚至与训练数据集大小相关的指数收敛。

这是一篇使用谱元方法与 PINN（物理信息神经网络）相结合的论文。

摘要​

摘要原文：

Spectral methods are an important part of scientific computing’s arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet) to identify candidate functions on which to expand the solution of PDEs. We have devised an approach that uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions that have the following properties: (1) they constitute a basis, (2) they are orthonormal, and (3) they are hierarchical, i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs. The proposed approach advances the state of the art and versatility of spectral methods and, more generally, promotes the synergy between traditional scientific computing and machine learning.

摘要翻译：

谱方法是科学计算用于求解偏微分方程 (PDE) 的重要工具。然而，它们的适用性和有效性在很大程度上取决于用于扩展偏微分方程解的基函数的选择。近十年来，深度学习异军突起，成为提供复杂函数高效表示的有力竞争者。在当前的工作中，我们提出了一种将深度神经网络与光谱方法相结合来求解 PDE 的方法。特别是，我们使用一种被称为深度算子网络（DeepONet）的深度学习技术来识别候选函数，并在此基础上扩展 PDE 的求解。我们设计了一种方法，以 DeepONet 提供的候选函数为起点，构建一组具有以下特性的函数：(1) 它们构成一个基础；(2) 它们是正交的；(3) 它们是分层的，即类似于傅里叶级数或正交多项式。我们利用定制基函数的有利特性，研究了它们的近似能力，并利用它们扩展了线性和非线性时变 PDE 的解。所提出的方法推进了频谱方法的技术水平和多功能性，更广泛地说，促进了传统科学计算与机器学习之间的协同作用。

其实只要细读了他的摘要就可以发现这篇论文的做法比较独特，他并不是把机器学习与谱方法相融合，而是采取了一种分阶段的模式，第一步是使用 DeepONet 去识别候选函数，再利用常规的谱方法来完成剩下的计算工作，他并不是一整块的模型而是耦合度低的分阶段模型。这种思想其实结合误差达到 $10^{-4}$ 到 $10^{-8}$ 就能够理解，不过也算是给机器学习融合谱方法提供了一种思路，接下来我们详细的阅读这篇文章的结果以及方法。