与多任务DeepONet的协同学习,以实现高效的PDE问题解决
Varun Kumar1, Somdatta Goswami2, Katiana Kontolati2
1School of Engineering, Brown University, United States of America.
概括
多任务学习 (MTL) 增强了通过部分微分方程 (PDEs) 控制的科学问题的神经网络概括性. 一个新的MT-DeepONet框架有效地解决了各种PDE任务,包括各种源术语和几何形状,降低了整体培训成本.
科学领域:
- 科学机器学习科学机器学习
- 计算科学与工程 计算科学与工程
- 神经操作员学习神经操作员学习
背景情况:
- 多任务学习 (MTL) 通过利用跨相关任务的信息来提高传统机器学习的概括性.
- 将MTL应用于由部分微分方程 (PDEs) 控制的科学问题是具有挑战性的,因为需要对特定任务进行修改.
- 现有的方法往往需要针对不同的物理过程或几何形状进行单独的培训.
研究的目的:
- 开发一个统一的框架来解决使用MTL在科学和工程中的各种PDE-governed问题.
- 为了提高神经运算符的概括能力,用于各种源条和几何学的问题.
- 为了降低与解决复杂的PDE任务相关的整体计算成本.
主要方法:
- 引入一个集成MTL原则的多任务深度运营商网络 (MT-DeepONet).
- 在DeepONet中修改分支网络,以处理PDEs中的参数化系数的各种功能形式.
- 在分支网络中包含二进制面具和损失术语来管理参数化的几何形状,改进融合和转移学习.
主要成果:
- 在三个基准问题上成功应用:使用不同源项的费舍尔方程,2D多重几何体的达西流和使用参数化几何体的3D热传输.
- 展示了改进的转移学习能力,以新的,未见的几何形状.
- 验证了MT-DeepONet能够预测新但相似的几何配置的解决方案的能力.
结论:
- MT-DeepONet框架为解决科学和工程领域广泛的PDE问题提供了一种新的统一方法.
- 通过MTL进行协同学习,大大降低了神经操作员的整体培训成本.
- 拟议的修改允许在单一培训范式内有效处理各种功能形式和几何形状.
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