用机器学习从数据中推断微分方程解的唯一性与稳定性。
Data-Driven, ML-assisted Approaches to Problem Well-Posedness
- 基于数据驱动方法，结合机器学习与流形学习推断方程解的存在唯一性。
- 在无严格数学证明条件下，仍能从观测数据中识别出良好适定性特征。
- 适合从事科学计算、物理模型构建的研究者参考。
经典方法求解微分方程需指定充分的初值或边界条件以确保解的存在性和唯一性。然而，实际数据采集常表现为在时空域任意位置观测到完整的解数据块，或获取同一算子生成的多个解。本文展示如何利用机器学习与流形学习工具，从数据中非参数化地推断微分方程问题的适定性特征，尤其针对现有理论无法保证存在唯一解的情形。研究融合数据同化与算子学习视角，实现对未知条件下解行为的可解释性分析。
原文摘要 · Abstract (English)
Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves studying the existence and uniqueness of solutions, and their dependence to such pre-specified conditions. However, in part due to mathematical necessity, these conditions are usually specified "to arbitrary precision" only on (appropriate portions of) the boundary of the space-time domain. This does not mirror how data acquisition is performed in realistic situations, where one may observe entire "patches" of solution data at arbitrary space-time locations; alternatively one might have access to more than one solutions stemming from the same differential operator. In our short work, we demonstrate how standard tools from machine and manifold learning can be used to infer, in a data driven manner, certain well-posedness features of differential equation problems, for initial/boundary condition combinations under which rigorous existence/uniqueness theorems are not known. Our study naturally combines a data assimilation perspective with an operator-learning one.
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