Data-Driven, ML-assisted Approaches to Problem Well-Posedness
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.
Code (0)
등록된 구현이 없습니다.
Tasks
Operator learningSimilar Papers 제목 키워드 기반
Optimization-driven Machine Learning for Intelligent Reflecting Surfaces Assisted Wireless Networks
Intelligent reflecting surface (IRS) has been recently employed to reshape the wireless channels by controlling individual scattering elements' phase shifts, namely, passive beamforming. Due to the large size of scatteri…
BIG-bench Machine LearningAnticipation-driven Adaptive Architecture for Assisted Living
Anticipatory expression underlies human performance. Medical conditions and, especially, aging result in diminished anticipatory action. In order to mitigate the loss, means for engaging still available resources (capabi…
Data-Driven Learning of Two-Stage Beamformers in Passive IRS-Assisted Systems with Inexact Oracles
We develop an efficient data-driven and model-free unsupervised learning algorithm for achieving fully passive intelligent reflective surface (IRS)-assisted optimal short/long-term beamforming in wireless communication n…
Noisy Labels are Treasure: Mean-Teacher-Assisted Confident Learning for Hepatic Vessel Segmentation
Manually segmenting the hepatic vessels from Computer Tomography (CT) is far more expertise-demanding and laborious than other structures due to the low-contrast and complex morphology of vessels, resulting in the extrem…
Evolving the pulmonary nodules diagnosis from classical approaches to deep learning aided decision support: three decades development course and future prospect
Lung cancer is the commonest cause of cancer deaths worldwide, and its mortality can be reduced significantly by performing early diagnosis and screening. Since the 1960s, driven by the pressing needs to accurately and e…