[论文解读] A fast and accurate domain-decomposition nonlinear manifold reduced order model

Alejandro N. Diaz, Youngsoo Choi+1 more|arXiv (Cornell University)|May 24, 2023

Model Reduction and Neural Networks参考文献 70被引用 9

一句话总结

这项工作将非线性流形ROM与领域分解相结合，使用宽而浅的稀疏自编码器和超降维来并行训练子域NM-ROM，并实现比 DD LS-ROM 在 2D Burgers 方程上更快、更高精度的解。

ABSTRACT

This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM-ROMs approximate the full order model (FOM) state in a nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for problems with slowly decaying Kolmogorov n-width. However, the number of NM-ROM parameters that need to be trained scales with the size of the FOM. Moreover, for "extreme-scale" problems, the storage of high-dimensional FOM snapshots alone can make ROM training expensive. To alleviate the training cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and couples them to obtain a global NM-ROM. This approach has several advantages: Subdomain NM-ROMs can be trained in parallel, involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and can be tailored to subdomain-specific features of the FOM. The shallow, sparse architecture of the autoencoder used in each subdomain NM-ROM allows application of hyper-reduction (HR), reducing the complexity caused by nonlinearity and yielding computational speedup of the NM-ROM. This paper provides the first application of NM-ROM (with HR) to a DD problem. In particular, this paper details an algebraic DD reformulation of the FOM, training a NM-ROM with HR for each subdomain, and a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and a priori and a posteriori error estimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM with HR on the 2D steady-state Burgers' equation, showing an order of magnitude improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM.

研究动机与目标

• 以领域分解为手段，降低离线训练成本并提高大尺度FOMs上NM-ROM的可扩展性。
• 为每个子域开发带有超降维的DD NM-ROM，并将它们耦合以形成全局NM-ROM。
• 为DD NM-ROM with HR提供理论收敛性结果和误差估计。
• 在2D稳态Burgers方程上演示该方法，并与带HR的DD LS-ROM进行比较。

提出的方法

• 将FOM的代数域分解表述为带端口兼容性约束的约束非线性最小二乘问题。
• 使用宽而浅的稀疏自编码器提供的低维映射来近似子域状态的 NM-ROM。
• 应用超降维以降低每个子域 NM-ROM 中非线性项的计算成本。
• 强/弱ROM端口约束以通过端口耦合子域并获得全局降阶系统。
• 用不完全的Lagrange-Newton SQP方法求解得到的约束LSPG-ROM，并给出收敛性分析。

实验结果

研究问题

• RQ1 将域分解(DD) 集成的 NM-ROM 能否在降低每个子域的训练数据和参数数量的同时实现全局解的高精度？
• RQ2 超降维是否在 DD NM-ROMs 中实现计算加速，且速度提升可与 DD LS-ROM 相媲美或更好，同时保持精度？
• RQ3 强/弱 ROM-端口约束如何影响耦合精度和求解器的收敛性？
• RQ4 在此 DD 设置中 DD NM-ROM with HR 的先验和后验误差界限有哪些？

主要发现

• 带有 HR 的 DD NM-ROM 在两维稳态 Burgers 方程上相较于带 HR 的 DD LS-ROM，在精度上实现了数量级级别的提升。
• 子域 NM-ROM 可以并行训练，且需要更少的子域训练数据，降低离线成本。
• 宽、浅、稀疏自编码器架构实现了有效的超降维，对非线性项显著加速。
• 该框架包含一个在理论上有依据的不完全 Lagrange-Newton SQP 求解器，并给出收敛性结果。
• 该方法将带 HR 的 NM-ROM 推广到 DD 设置，并为 DD NM-ROM with HR 提供先验与后验误差估计。