Partial Differential Equations in the Age of Machine Learning: A Critical Synthesis of Classical, Machine Learning, and Hybrid Methods
Partial differential equations (PDEs) govern physical phenomena across the full range of scientific scales, yet their computational solution remains one of the defining challenges of modern science. This critical review examines two mature but epistemologically distinct paradigms for PDE solution, classical numerical methods and machine learning approaches, through a unified evaluative framework organized around six fundamental computational challenges. Classical methods are assessed for their structure-preserving properties, rigorous convergence theory, and scalable solver design; their persistent limitations in high-dimensional and geometrically complex settings are characterized precis...
| Authors | Mohammad Nooraiepour, Jakub Wiktor Both, Teeratorn Kadeethum et al. |
|---|---|
| Primary category | cs.LG |
| Published | 8 March 2026 |
| Last updated at source | 8 March 2026 |
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