BEACONS：具有误差界限的代数可组合神经求解器用于偏微分方程
BEACONS: Bounded-Error, Algebraically-Composable Neural Solvers for Partial Differential Equations
AI总结：
BEACONS通过构建具有严格误差界限的代数可组合神经求解器，解决PDE在远超训练数据范围的可靠求解问题。
摘要：
传统神经网络在可靠地泛化到其训练数据凸包之外的局限性在计算物理学中构成了一个重大问题，因为在其中常常希望在远超任何可以实验或解析验证的领域求解偏微分方程（PDEs）。在本文中，我们展示了如何通过构建形式上验证的PDE神经求解器来克服这些限制，这些求解器具有严格收敛、稳定性和守恒性质，因此即使在外推领域也能保证正确性。通过使用特征方法预测PDE解的分析性质（即使在训练域任意远的区域），我们展示了如何构造严格外推界限，以限制浅层神经网络近似中的最坏L^∞误差。然后，通过将PDE解分解为更简单的函数的组合，我们展示了如何基于组合深度学习的想法将这些浅层神经网络组合成深度架构，在其中近似中的大L^∞误差已被抑制。所得到的框架称为BEACONS（有界误差、代数可组合神经求解器），它包括用于自身神经求解器的自动代码生成器以及定制的自动定理证明系统，用于生成机器可验证的正确性证书。我们将其应用于各种线性和非线性PDEs，包括线性输运和无粘 Burgers 方程以及完整的可压缩欧拉方程，分别在1D和2D中，并展示了BEACONS架构如何能够可靠且有界地在训练数据之外外推解。讨论了该方法相对于经典PINN方法的各种优势。
英文摘要：
The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond anything which can be experimentally or analytically validated. In this paper, we show how it is possible to circumvent these limitations by constructing formally-verified neural network solvers for PDEs, with rigorous convergence, stability, and conservation properties, whose correctness can therefore be guaranteed even in extrapolatory regimes. By using the method of characteristics to predict the analytical properties of PDE solutions a priori (even in regions arbitrarily far from the training domain), we show how it is possible to construct rigorous extrapolatory bounds on the worst-case L^inf errors of shallow neural network approximations. Then, by decomposing PDE solutions into compositions of simpler functions, we show how it is possible to compose these shallow neural networks together to form deep architectures, based on ideas from compositional deep learning, in which the large L^inf errors in the approximations have been suppressed. The resulting framework, called BEACONS (Bounded-Error, Algebraically-COmposable Neural Solvers), comprises both an automatic code-generator for the neural solvers themselves, as well as a bespoke automated theorem-proving system for producing machine-checkable certificates of correctness. We apply the framework to a variety of linear and non-linear PDEs, including the linear advection and inviscid Burgers' equations, as well as the full compressible Euler equations, in both 1D and 2D, and illustrate how BEACONS architectures are able to extrapolate solutions far beyond the training data in a reliable and bounded way. Various advantages of the approach over the classical PINN approach are discussed.
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