非线性梯度流方程中非局部相互作用核的稀疏识别：基于部分反演
Sparse identification of nonlocal interaction kernels in nonlinear gradient flow equations via partial inversion
AI总结：
本文研究从含噪轨迹数据中识别非线性聚集-扩散方程的非局部相互作用势，提出通过最小化二次误差泛函并辅以稀疏正则化构建变分问题，采用部分反演算法求解，并给出PDE稳定性估计与误差分析，一维与二维示例验证了方法的有效性。
摘要：
我们探讨了从含噪声的离散轨迹数据中识别非线性聚集-扩散方程里的非局部相互作用势的反问题。我们的方法构建并求解了一个正则化变分问题，该问题要求在一组假设函数上最小化二次误差泛函，并进一步通过增强稀疏性的正则化项进行扩展。我们采用一种类似于CoSaMP [57]和子空间追踪算法[31]的部分反演算法来求解基追踪问题。一项关键的理论贡献是我们针对PDE提出的全新稳定性估计，验证了误差泛函在控制由真实和估计的相互作用势生成的解之间的2-Wasserstein距离方面的能力。我们的工作还包括对实际实现中由离散化和观测误差引起的估计量误差的分析。我们通过各种展示集体行为的一维和二维示例证明了该方法的有效性。
英文摘要：
We address the inverse problem of identifying nonlocal interaction potentials in nonlinear aggregation-diffusion equations from noisy discrete trajectory data. Our approach involves formulating and solving a regularized variational problem, which requires minimizing a quadratic error functional across a set of hypothesis functions, further augmented by a sparsity-enhancing regularizer. We employ a partial inversion algorithm, akin to the CoSaMP [57] and subspace pursuit algorithms [31], to solve the Basis Pursuit problem. A key theoretical contribution is our novel stability estimate for the PDEs, validating the error functional ability in controlling the 2-Wasserstein distance between solutions generated using the true and estimated interaction potentials. Our work also includes an error analysis of estimators caused by discretization and observational errors in practical implementations. We demonstrate the effectiveness of the methods through various 1D and 2D examples showcasing collective behaviors.
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