HomPINNs:用于解决多个解决方案的非线性微分方程的反向问题的同位素物理信息的神经网络
Haoyang Zheng1, Yao Huang2, Ziyang Huang3
1School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, USA.
同位体物理信息神经网络 (HomPINNs) 解决了多个解决方案的非线性微分方程的复杂反向问题. 这种新的框架有效地识别了科学计算应用中的各种解决方案和未知的参数.
科学领域:
- 计算数学 计算数学 计算数学
- 科学计算科学计算
- 应用物理 应用物理
背景情况:
- 解决非线性微分方程 (DE) 的反向问题是具有挑战性的,因为由于非唯一性,对称性和分叉而产生的多个解决方案.
- 现有的方法很难有效地识别和描述复杂系统中的各种解决方案空间.
研究的目的:
- 引入一个新的框架,同位体物理信息的神经网络 (HomPINNs),用于解决多个解决方案的非线性DE的反向问题.
- 展示HomPINNs在识别未知的参数和各种解决方案方面的可扩展性和适应性.
主要方法:
- 利用神经网络 (NN) 在执行DE约束的同时,在多个解决方案中对未标记的观察结果进行近似计算.
- 采用同位素延续来追踪观测并系统地识别不同的解决方案.
- 在一维DE和二维Gray-Scott模拟上测试HomPINNs.
主要成果:
- 对于测试的非线性DE,HomPINNs成功地确定了多个解决方案和未知的参数.
- 该框架展示了跨不同维度问题的可扩展性和适应性.
- 在二维Gray-Scott模拟上的验证证实了该方法的实际适用性.
结论:
- 同位体物理信息的神经网络为解决具有挑战性的反向问题提供了有效的解决方案,涉及非线性DE的多个解决方案.
- 提出的方法为推进科学计算提供了巨大的潜力,特别是在各种科学学科的复杂系统建模方面.
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