This paper investigates the application of regularized neural networks (RNNs) to the numerical solution of stochastic differential equations (SDEs), forward-backward stochastic differential equations (FBSDEs), and partial differential equations (PDEs). A novel regularization framework that incorporates discretization errors and \(L^1\) penalties into the loss function is proposed, enabling robust learning of complex stochastic dynamics. For SDEs, numerical experiments on geometric Brownian motion demonstrate that our method achieves a significantly lower mean square error (MSE = 0.0786) compared to the Euler-Maruyama scheme (MSE = 0.1246), showing superior accuracy and convergence. In solving FBSDEs, the problem is reformulated as a stochastic optimal control task and solved via regularized neural networks, leveraging the equivalence between FBSDEs and control problems. This approach ensures solution existence and uniqueness while maintaining computational flexibility. For PDEs, stochastic processes and deterministic equations are bridged using the Feynman-Kac formula, and our model is validated on the Allen-Cahn equation. Results show that the relative error in \(\hat{Y}_0\) sharply drops from 142.68% to 0.018% as training progresses. Our framework offers a scalable, adaptive alternative to traditional numerical solvers, especially in high-dimensional or nonlinear settings. These results underscore the potential of regularized deep learning in solving a broad class of stochastic and deterministic problems efficiently.