Problem
Research questions and friction points this paper is trying to address.
partial differential equations
physics-informed neural networks
robustness
extensibility
point-to-point fitting
General Explicit Network (GEN): A novel deep learning architecture for solving partial differential equations
🤖 AI Summary
Traditional physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by pointwise fitting, which neglects the global structure of the solution and consequently limits generalization and robustness. This work proposes the Generalized Explicit Network (GEN), which introduces a novel "point-to-function" paradigm for PDE solving. By incorporating physical priors to construct basis functions, GEN explicitly models the solution space, replacing the conventional strategy of local activation functions. This approach substantially enhances model scalability, robustness, and generalization capability, consistently outperforming existing methods across a variety of PDEs.
📝 Abstract
Machine learning, especially physics-informed neural networks (PINNs) and their neural network variants, has been widely used to solve problems involving partial differential equations (PDEs). The successful deployment of such methods beyond academic research remains limited. For example, PINN methods primarily consider discrete point-to-point fitting and fail to account for the potential properties of real solutions. The adoption of continuous activation functions in these approaches leads to local characteristics that align with the equation solutions while resulting in poor extensibility and robustness. A general explicit network (GEN) that implements point-to-function PDE solving is proposed in this paper. The "function" component can be constructed based on our prior knowledge of the original PDEs through corresponding basis functions for fitting. The experimental results demonstrate that this approach enables solutions with high robustness and strong extensibility to be obtained.
Innovation
Methods, ideas, or system contributions that make the work stand out.
General Explicit Network
point-to-function learning
physics-informed neural networks
basis function representation
partial differential equations
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