基于卷积算子压缩的非线性物理降阶模型
A Nonlinear Physics-based Reduced Order Model with Convolutional-based Operator Compression
- SISSA, International School for Advanced Studies（国际高等研究院）
- Sant’Anna School of Advanced Studies（圣安娜高等学院）
- The Biorobotics Institute（生物机器人研究所）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出一种非线性物理降阶模型，通过卷积架构压缩全阶算子而非仅压缩解场，结合径向基函数插值实现高效在线预测，在三个参数化问题上验证了其有效性。
中文摘要 AI 辅助
降阶模型（ROMs）被广泛用于加速参数化偏微分方程的求解，其中基于投影的方法依赖于在低维空间中对控制方程进行线性表示。然而，其效率从根本上受到解流形的Kolmogorov N宽度的限制：当N宽度衰减缓慢时，精确逼近需要大量的线性模态。这一限制促使了非线性降阶模型的发展，这类模型通常通过自编码器架构学习到的非线性潜在表示来替代解场的线性表示。在本工作中，我们提出了一种不同的视角，即将非线性直接引入控制算子的表示中。我们不是仅压缩解场，而是学习全阶算子的非线性低维表示，同时保留显式的降阶方程组。采用卷积架构以利用微分算子的局部性和空间结构，所得的压缩算子通过一种同时考虑重构精度和降阶解误差的训练策略与解的非线性表示耦合。为了处理由精细和非结构化离散化产生的大型稀疏算子，使用连续卷积直接对其非零元素进行操作。最后，径向基函数插值能够在未见过的参数配置下预测压缩算子，从而在不组装相应全阶系统的情况下提供高效的在线阶段。在三个参数化问题上的数值实验证明了非线性算子压缩作为基于方程的降阶模型的一种替代策略的潜力。
英文摘要
Reduced-order models (ROMs) are widely used to accelerate the solution of parametrized partial differential equations, with projection-based methods relying on a linear representation of the governing equations in a low-dimensional space. Their efficiency, however, is fundamentally limited by the Kolmogorov N-width of the solution manifold: when the N-width decays slowly, accurate approximations require a large number of linear modes. This limitation has motivated nonlinear ROMs, which typically replace the linear representation of the solution field with nonlinear latent representations learned through autoencoder architectures. In this work, we propose a different perspective by introducing nonlinearity directly into the representation of the governing operators. Rather than compressing only the solution field, we learn nonlinear low-dimensional representations of the full-order operators while retaining an explicit reduced system of equations. Convolutional architectures are employed to exploit the locality and spatial structure of differential operators, and the resulting compressed operators are coupled with a nonlinear representation of the solution through a training strategy that accounts for both reconstruction accuracy and the error of the reduced-order solution. To address large sparse operators arising from fine and unstructured discretizations, continuous convolutions are used to operate directly on their non-zero entries. Finally, radial basis function interpolation enables the prediction of the compressed operators at unseen parameter configurations, providing an efficient online stage without assembling the corresponding full-order system. Numerical experiments on three parametrized problems demonstrate the potential of nonlinear operator compression as an alternative strategy for equation-based ROMs.
