General Explicit Network (GEN): A Novel Deep Learning Architecture for Solving Partial Differential Equations
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.
Introduction
The application of deep learning techniques to solve partial differential equations has garnered significant attention in recent years. Traditional numerical methods often struggle with complex geometries and boundary conditions, making them less effective in certain scenarios. The introduction of PINNs has offered a promising alternative; however, limitations persist in their practical application.
Limitations of Existing Methods
While PINNs have shown potential, they primarily operate under the following constraints:
- Discrete Point-to-Point Fitting: Many existing methods focus on fitting specific points rather than capturing the continuous nature of solutions.
- Poor Extensibility: The reliance on local characteristics often limits the ability to generalize to a broader range of problems.
- Robustness Issues: Continuous activation functions can lead to instability in solutions, particularly in complex scenarios.
Introducing the General Explicit Network (GEN)
The General Explicit Network (GEN) represents a significant advancement in addressing the shortcomings of traditional PINNs. By employing a point-to-function approach, GEN facilitates a more holistic understanding of PDE solutions. This method allows for the construction of the “function” component using prior knowledge of the original PDEs, which can be modeled through appropriate basis functions.
Key Features of GEN
- Point-to-Function Implementation: Unlike traditional methods, GEN captures the entire function space, leading to more accurate and reliable solutions.
- Basis Function Utilization: The incorporation of basis functions derived from prior knowledge enhances the fitting process and improves overall performance.
- High Robustness and Extensibility: Experimental results have demonstrated that GEN can adapt to various problems, showcasing its ability to maintain solution integrity across different scenarios.
Conclusion
The General Explicit Network (GEN) marks a notable evolution in the application of deep learning for solving partial differential equations. By addressing the limitations of existing methods and leveraging prior knowledge through basis functions, GEN offers a promising path forward. Future research and development will likely explore the full potential of this architecture in diverse applications, paving the way for broader deployment in both academic and industrial settings.
References
For further details, refer to the original paper available on arXiv: 2604.03321v1.