UC Santa Barbara

Partial differential equations (PDEs) are central to modeling physical, biological, and engineered systems, yet their analytical solutions are rare, and numerical methods often face scalability and efficiency challenges in high dimensions or complex domains. Recent advances in deep learning have led to the development of neural network-based PDE solvers, including physics-informed neural networks (PINNs), the Deep Ritz and Deep Galerkin methods, and neural operator frameworks such as Fourier Neural Operators (FNOs) and DeepONets. These approaches offer mesh-free, differentiable alternatives to classical solvers and demonstrate promise in handling high-dimensional, multi-physics, and data-sparse problems.This survey presents a comprehensive review of deep neural network (DNN)-based PDE solvers, beginning with foundational variational and residual-based formulations, and progressing to operator learning approaches that generalize across input function spaces. We compare algorithmic frameworks, training paradigms, and performance characteristics, and highlight applications across physics, engineering, and biology. Key implementation challenges—such as enforcing boundary conditions, ensuring convergence, and scaling to high-dimensional settings—are discussed alongside recent strategies to mitigate them. Finally, we explore emerging directions, including hybrid solvers, uncertainty quantification, and scientific foundation models. This survey aims to provide researchers and practitioners with a structured overview of the field and guidance for future development of scalable, data-integrated, and physically consistent PDE solvers.