基于神经矩阵算子的实时逆问题求解
Real-time inverse solutions via neural matrix operators
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中文总结 AI 辅助
本研究提出NEMO方法，可实时求解受PDE约束的逆问题，速度较传统方法提升超三个数量级，性能与最先进多输入神经算子相当且在线复杂度更低，还支持实时不确定性量化，应用于污染物迁移和高超声速载荷识别场景。
中文摘要 AI 辅助
数字孪生中的实时预测与控制需要快速数据同化。对于许多物理系统，数据同化任务需要求解受物理约束的逆问题，而使用传统物理求解器在实时场景下求解该问题往往计算上难以处理。本研究提出一种降阶基神经算子方法，用于在数字孪生场景下实现逆问题的实时求解。该方法专门针对一类由偏微分方程（PDE）支配的时空动力学问题，这类问题关于模型参数$m$呈非线性参数化，关于反演参数$q$呈线性参数化。基于该物理结构，本研究的神经算子在降阶子空间中近似从模型参数$m$到可观测量算子$\boldsymbol{\textit{F}}(m)$的非线性映射。由于该神经算子的输出是可观测量算子本身（表现为矩阵形式），本研究将该方法命名为神经矩阵算子（NEural Matrix Operator, NEMO）。借助NEMO，对于新给定的$m$，可在降阶子空间中实现$q$的闭式逆问题求解。本研究将NEMO应用于两个实际场景：污染物迁移初始条件识别与高超声速飞行器载荷识别。结果表明，NEMO可实现高质量的逆问题求解，满足数据同化的实时需求，与使用PDE求解器构建降阶算子的方式相比，速度提升超过三个数量级。此外，NEMO的逆问题求解性能与最先进的多输入神经算子相当，同时将在线计算复杂度降低超过一个数量级，还可提供实时不确定性量化。
英文摘要
Rapid data assimilation is required for real-time prediction and control in digital twins. For many physical systems, the data assimilation task requires the solution of a physics-constrained inverse problem, which is often computationally intractable in real time using traditional physics solvers. This work presents a reduced-basis neural operator approach to enable real-time inverse problem solutions in the digital twin setting. Our approach specifically targets the large class of problems with spatiotemporal dynamics governed by partial differential equations (PDEs) that are parameterized nonlinearly with respect to model parameters $m$, and linearly with respect to inversion parameters $q$. Based on this physical structure, our neural operator approximates the nonlinear map from the model parameters $m$ to the parameter-to-observable operator $\mathcal{F}(m)$ in a reduced subspace. Since the output of the neural operator is the parameter-to-observable operator itself (manifested as a matrix), we refer to this approach as NEural Matrix Operator (NEMO). With NEMO, for new given $m$, we enable a closed-form inverse problem solution for $q$ in a reduced subspace. We apply NEMO in two real-world applications: contaminant transport initial condition identification, and hypersonic vehicle load identification. We show that NEMO delivers high quality inverse problem solutions for data assimilation in real time, with over three orders of magnitude speedup compared to constructing the reduced operator with the PDE solver. Further, NEMO demonstrates comparable inverse performance to a state-of-the-art multiple-input neural operator, while reducing online computational complexity by over an order of magnitude and providing real-time uncertainty quantification.
