Learning data-driven discretizations for partial differential equations
Learning data-driven discretizations for partial differential equations
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DOI：
10.1073/pnas.1814058116
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发表时间：
2019-07-30
影响因子：
11.1
通讯作者：
Brenner, Michael P.
中科院分区：
文献类型：
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作者：
Bar-Sinai, Yohai;Hoyer, Stephan;Brenner, Michael P.
偏微分方程组(PDE)的数值解具有挑战性，因为需要在很大的长度和时间尺度上解析时空特征。通常，解析解决方案中最精细的特征在计算上是困难的。唯一的办法是使用近似的粗粒度表示，其目的是准确地表示长波长动力学，同时适当地考虑未解决的小规模物理。推导这样的粗粒度方程是出了名的困难，而且往往是即兴的。在这里，我们引入数据驱动离散化，这是一种基于已知基本方程的实际解来学习偏微分方程组的优化近似的方法。我们的方法使用神经网络来估计空间导数，并对其进行端到端的优化，以最大限度地满足低分辨率网格上的方程。由此产生的数值方法是非常精确的，使我们能够以比标准有限差分方法可能的4x到8x粗的分辨率及时地积分一维空间中的非线性方程组。
The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4x to 8x coarser than is possible with standard finite-difference methods.