用物理信息的神经网络解决维度的诅咒
Zheyuan Hu1, Khemraj Shukla2, George Em Karniadakis2
1National University of Singapore, 21 Lower Kent Ridge Road, 119077, Singapore.
概括
一种名为"随机维度梯度下降" (SDGD) 的新方法使得基于物理学的神经网络 (PINNs) 能够在单个GPU上有效地解决高维部分微分方程 (PDEs),克服维度的诅咒.
科学领域:
- 计算数学 计算数学 计算数学
- 机器学习用于科学
- 高维建模 高维建模
背景情况:
- 维度的诅咒显著增加了解决高维部分微分方程 (PDEs) 的计算成本.
- 对于高维 PDE 的现有方法在计算上昂贵,并且对于一般的非线性问题没有实现真正的缩放.
- 基于物理学的神经网络 (PINNs) 提供了一个无网格的方法,但在可扩展到非常高的维度方面存在困难.
研究的目的:
- 开发一种用于缩放物理信息神经网络 (PINNs) 的新方法,以解决任意高维 PDEs.
- 在解决复杂的PDEs时,解决维度的诅咒所带来的计算挑战.
主要方法:
- 引入了静态维度梯度下降 (SDGD),这是PINN的新培训方法.
- SDGD将PDE和PINN残余梯度分解为维度组件,并每次代取样这些维度的子集.
- 理论上的收和SDGD的属性已被证明.
主要成果:
- SDGD使PINNs能够解决众所周知的困难的高维PDEs,包括汉密尔顿-雅各比-贝尔曼 (HJB) 和施罗丁格方程,在数万个维度.
- 在单个GPU上演示了快速解决方案,在1小时内解决1000维非线性PDEs,在12小时内解决10万维的PDEs.
- 通过复杂的,异构的和不可分割的溶液成功地解决了非线性PDEs.
结论:
- SDGD是一种可通用的培训方法,有效地为任意的高维PDEs扩展PINN.
- 该方法克服了维度的诅咒,在标准硬件上提供快速高效的解决方案.
- SDGD可以与当前和未来的PINN变体集成,以解决以前难以解决的高维问题.
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