Deep Learning Acceleration of Numerical Solutions for Nonlinear Time-Dependent Partial Differential Equations

Abstract

The rapid advancement of deep learning has significantly transformed the numerical solution of nonlinear time-dependent partial differential equations (PDEs), offering powerful alternatives to conventional computational methods such as Finite Difference Methods (FDM), Finite Element Methods (FEM), and spectral techniques. Traditional solvers often suffer from the curse of dimensionality, mesh dependency, and computational inefficiency when applied to highly nonlinear or high-dimensional systems. This study examines the emerging role of deep learning architectures in accelerating PDE solutions through mesh-free, data-driven, and physics-informed frameworks. Particular attention is given to Physics-Informed Neural Networks (PINNs), Deep Galerkin Methods (DGM), Deep Ritz Methods (DRM), Time-Induced Neural Networks (TINN), Physics-Informed Autoregressive Networks (PIANO), and neural operator approaches such as Fourier Neural Operators (FNO) and DeepONet. The study explores how these methods incorporate physical laws directly into neural network training through residual minimization, variational principles, and operator learning. Advanced temporal strategies including autoregressive modeling, domain decomposition, and parareal parallelization are analyzed for improving stability and long-term prediction accuracy in nonlinear dynamical systems. The review also discusses uncertainty quantification using Gaussian Processes, dimensionality reduction techniques, and the integration of Neural Ordinary Differential Equations (NODEs) for efficient latent-space evolution. Benchmarking frameworks such as PDEBench are evaluated to compare solver accuracy, scalability, and physical consistency across diverse PDE classes. Despite significant progress, challenges related to interpretability, training stability, sharp gradient reconstruction, and computational reliability remain active research areas. Overall, deep learning-based PDE solvers represent a transformative paradigm in scientific computing, enabling faster, scalable, and highly adaptable solutions for complex systems in physics, engineering, finance, and computational science.