[论文解读] A Causality-DeepONet for Causal Responses of Linear Dynamical Systems

Lizuo Liu, Kamaljyoti Nath+1 more|arXiv (Cornell University)|Sep 17, 2022

Structural Health Monitoring Techniques被引用 8

一句话总结

本文提出 Causality-DeepONet，一种具备内置因果结构与卷积权重结构的深度算子网络，可精确学习线性动力系统中的因果响应算子，尤其适用于建筑物的地震响应。通过将物理因果性与时变齐次动力学编码进网络架构，该方法在仅使用极少训练数据的情况下实现了高精度，即使仅用10组训练样本，其性能也优于标准 DeepONet 变体。

ABSTRACT

In this paper, we propose a DeepONet structure with causality to represent the causal linear operators between Banach spaces of time-dependent signals. The theorem of universal approximations to nonlinear operators proposed in \cite{tianpingchen1995} is extended to operators with causalities, and the proposed Causality-DeepONet implements the physical causality in its framework. The proposed Causality-DeepONet considers causality (the state of the system at the current time is not affected by that of the future, but only by its current state and past history) and uses a convolution-type weight in its design. To demonstrate its effectiveness in handling the causal response of a physical system, the Causality-DeepONet is applied to learn the operator representing the response of a building due to earthquake ground accelerations. Extensive numerical tests and comparisons with some existing variants of DeepONet are carried out, and the Causality-DeepONet clearly shows its unique capability to learn the retarded dynamic responses of the seismic response operator with good accuracy.

研究动机与目标

• 为解决在地震激励下结构动力学中学习因果响应算子的挑战。
• 将物理因果性——即当前系统状态仅依赖于过去和当前输入——整合进 DeepONet 框架。
• 通过将滞后格林函数的时间齐次、卷积结构编码进网络权重，提升泛化能力并减少数据需求。
• 在仅使用少量训练数据的情况下，展示 Causality-DeepONet 在预测建筑物对地震地面加速度响应方面优于标准 DeepONet 变体。

提出的方法

• Causality-DeepONet 将非线性算子的通用逼近定理扩展至包含巴拿赫空间中时间相关信号之间的因果算子。
• 通过将支路网络中的标准全连接权重替换为依赖于输入与输出之间时间延迟的卷积类权重，强制实现因果性。
• 网络使用主干网络编码时间基函数，支路网络将输入信号（如地面加速度）映射到潜在空间。
• 输出通过主干网络输出的加权和计算得出，其中权重由支路网络确定，并被设计为反映卷积、时不变核的结构。
• 通过在时间延迟上共享卷积类权重，架构中嵌入了时间齐次性这一物理特性，模仿滞后格林函数的结构。
• 采用自定义损失函数进行训练，结合 L2 正则化与学习率调度，以防止过拟合，尤其在小样本训练时至关重要。

Figure 1: A typical ground acceleration of due to earthquake: The ground acceleration is due to 14383980 earthquake recorded at station North Hollywood, 2008 (a) time history of the acceleration (b) frequency spectrum. The absolute maximum acceleration is indicated which is also known as Peak Ground

实验结果

研究问题

• RQ1 能否系统性地修改 DeepONet 架构，以在学习线性动力系统算子时强制实现物理因果性？
• RQ2 在学习因果响应算子时，引入卷积权重结构如何提升泛化能力并减少数据需求？
• RQ3 Causality-DeepONet 是否能在远少于标准 DeepONet 变体的训练样本下，实现对建筑地震响应的高精度预测？
• RQ4 时间齐次、卷积权重设计的引入在多大程度上增强了模型对未见地震输入的泛化能力？

主要发现

• 在仅使用10组训练样本时，Causality-DeepONet 的相对 L2 误差显著低于标准 DeepONet 变体，证明其具有卓越的数据效率。
• 不带卷积结构的变体即使使用100组训练样本也未能达到满意的精度，凸显卷积结构在有效学习中的必要性。
• 具备卷积与因果性约束的 Causality-DeepONet 在地震激励下对建筑物屋顶位移响应的预测中表现出良好精度。
• 该网络在不同激活函数与网络规模下均保持高性能，表明其对网络超参数具有鲁棒性。
• 因果性与卷积权重设计的结合，使模型能够准确学习滞后格林函数，该函数可建模结构对过去和当前地震输入的物理响应。
• 在测试案例中，该方法优于标准 DeepONet 与多尺度 DeepONet，证实其在因果性、振荡性动力系统中的有效性。

Figure 2: Schematic Diagram of the DeepONet: A schematic diagram of DeepONet showing branch and the trunk net along with the input data and output. The number of the input neurons of the branch net is equal to the number of sensor points in the input signals. The trunk net takes the input point $x$