From Differential Equations to Deep Networks: A Unified Applied Mathematics and Computer Science Framework for Physics-Informed Computational Mechanics

Here’s a line that’s been true for a while now but that we don’t talk about enough: the oldwalls between pure mathematical analysis, numerical computation, and mechanical modelingare quietly coming down, and modern scientific machine learning is basically the wreckingball. In this paper I try to pull together an account of five closely related research specialties– Mathematics, Applied Mathematics and Computer Science, Mechanics and Mathematical Modeling, Applied Mathematics, and Mathematics and Computer Science– and showthey’re all really orbiting one shared object: the numerical and data-driven solution of differential equations that govern mechanical and physical systems. I went through more thanfifty peer-reviewed and archival sources, published between 1956 and 2024, and traced a path– classical variational and finite-element methods, then sparse-regression equation discovery,then physics-informed neural networks, then neural operators. From that I built a four-layerframework: mathematical formulation, discretization and algorithmic design, computationallearning, mechanical application. Then I compared method families across accuracy, computational cost, data efficiency, and interpretability, because a comparison without teeth isn’tworth much. To back all this up properly I’ve included explicit mathematical formulations,convergence bounds, functional-analytic loss landscapes, and three architectural diagramsthat lay the whole pipeline out visually– partly for the reader, honestly, and partly becausedrawing it out is how I convinced myself the four-layer split actually holds up. What fallsout of all this is a gap that just won’t close on its own: theoretical convergence guarantees, generalization behavior, and how (or whether) any of this gets taught in a coherentway remain seriously underdeveloped for hybrid mathematics–computer-science–mechanicsmethods. I end with concrete objectives and research questions aimed squarely at that gap,plus some thoughts on what it would take from higher-education curricula, national researchfunding, and interdisciplinary centers to actually close it. My hope, honestly, is that thisreads less like a conventional survey and more like something an early-career researcher ordoctoral candidate could keep open in a tab while figuring out where their own work actuallysits across these five specialties.