Deep neural networks (DNNs) approximate Partial Differential Equations (PDEs) in high-dimensional domains and complex boundary conditions that numerical methods struggle with this study. DNNs, tensor-product spaces, and sparse grids solve multiple-variable PDE problems' curse of dimensionality. Convergence, error boundaries, and the usefulness of depth vs. width network designs in managing high-dimensional inputs are examined in theory and experiment. The study utilized multiple datasets, each corresponding to different well-known partial differential equations (PDEs), such as the Navier-Stokes Equation, Schrödinger Equation, Hamilton-Jacobi Equation, Poisson Equation, Heat Equation, and Burgers' Equation. DNN training is examined using energy functional minimization and classical variational concepts to get insight into PDE solution regularity, existence, and uniqueness. Although convergence proofs and error estimates are still being established, DNNs can approximate quantum physics, fluid dynamics, and financial modelling PDEs. The study found that Latin hypercube and quasi-random sampling reduce high-dimensional DNN training costs and maintain accuracy. It also proposes studying DNN-based solver convergence guarantees and hybrid analytical-numerical methods that combine old solvers with new machine learning methods because real success differs from theoretical models. This extensive study shows that DNNs can solve complex, multidimensional PDEs and enriches the theoretical framework with high-dimensional function approximation, convergence features, and optimal network topologies. The findings suggest studying DNNs for stochastic PDEs with uncertainty and dimensional complexity. PDE estimates may be more accurate and efficient with classical and DNN-based hybrid approaches. This study promotes machine learning in pure and applied mathematics and creative solutions to old mathematical problems using empirical observations and strong mathematical theory.