Data-Driven, ML-assisted Approaches to Problem Well-Posedness

2025-03-25

AI Summary

This work addresses initial/boundary-value problems for differential equations lacking rigorous existence-uniqueness theory. We propose a data-driven framework for well-posedness assessment, diverging from classical theoretical analysis that relies on exact boundary conditions. Instead, our method leverages sparse, non-boundary, and multi-solution observational data—common in real-world measurements—and introduces the first integration of data assimilation with implicit operator learning. By synergistically combining manifold learning, supervised and self-supervised learning, and uncertainty-aware regression, the framework enables learnable inference of existence, uniqueness, and stability of solutions. Evaluated across multiple canonical PDE settings, it successfully identifies well-posed regimes and reduces estimation error by 37% compared to conventional heuristic criteria. Results demonstrate the feasibility, robustness, and generalizability of data-driven well-posedness evaluation.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

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.

Problem

• Infer well-posedness of differential equations using data-driven ML methods
• Analyze solution existence/uniqueness without traditional boundary conditions
• Combine data assimilation and operator-learning for PDE well-posedness

Innovation

• ML-assisted data-driven well-posedness inference
• Manifold learning for differential equation solutions
• Combining data assimilation with operator-learning