## 4 Conclusions  
In this paper, we proposed the HWPINN neural network architecture based on a wide-body structure and hard-constraint method, and we set up a series of numerical experiments using PINN to solve PDEs. Numerical experiments were designed using sparser PINN collection point densities in conjunction with defined boundary conditions, and the results show that HWPINN converges faster and makes more accurate predictions than SPINN. HWPINN is better than the PINN architecture with soft constraints in terms of the accuracy of the results and the speed of convergence. HWPINN still has disadvantages because it needs to be modeled with hard-constraint functions. Hard constraints also have the disadvantage of being inflexible and difficult to debug. Additionally, HWPINN does not perform consistently in dealing with some flexible boundary conditions. For example, when solving problems with Neumann boundaries, a residual function still needs to be construct to allow HWPINN to participate in fitting the constrained part of the process using traditional PINN technologies to increase its stability. In solving real physical problems, HWPINN needs a more refined design when facing complex boundary conditions. The flexibility and powerful approximation ability of traditional PINN enable it to handle all kinds of PDEs, whether homogeneous or nonhomogeneous, linear or nonlinear, systematic or monolithic; HWPINN discards the flexibility of PINN in boundary conditions to make the approximation smoother, converges more rapidly, is convenient for debugging, and provides better overall performance. When solving higher-dimensional problems, such as porous media flow, phase transition dynamics, and supercritical airfoil optimization, PINN is often confronted with the problem of being unable to debug and difficult to compute; then, the advantages of HWPINN, including ease of computation and smoothness of the fit, allow the modeling to be carried out normally. For the part of solving PDEs with which we were concerned, we set up several cases\" 1D AC equation, 1D Burgers equation, and 2D wave equation. The experimental results for different dimensional PDEs showed that HWPINN has better numerical stability; due to the more efficient network structure design, HWPINN can also converge to a suitable solution when the number of training iterations is small. For the traditional PINN, SPINN, and SWPINN with an added wide-body structure, the performance is not advantageous under the experimental conditions we set. The reason for this is that although PINN with soft constraints can more flexibly adapt to different problems, it requires more training iterations and more complex optimization strategies due to the complexity and instability of its constraints. In future work, we envision utilizing HWPINN to solve a wider range of PDE problems, with a special emphasis on utilizing HWPINN to solve positive and negative PDEs. In addition, we are keen to explore the synergies between HWPINN and physical models, with the intention of adopting this combined approach to efficiently solve PDEs in real-world problems. The potential applications in phase transition dynamics and CFD offer exciting prospects for pushing the limits of HWPINN's capabilities and refining its performance in a variety of complex physical scenarios. As our research progresses, our focus will extend beyond the traditional PDE cases explored in current experiments, and we will explore ways to improve the stability of HWPINN in solving real-world physics problems. We anticipate that incorporating HWPINN into the realm of phase transition dynamics and CFD will not only enhance its versatility and provide solutions for high-dimensional PDEs but will also provide groundbreaking solutions to real-world problems involving complex physical processes. Conceptualization, S.C. and Z.L. ; data curation, W.Z. ; investigation, Z.L., W.Z. and J.Y. ; methodology, S.C. and Z.L. ; project administration, W.Z. and J.Y. ; resources, S.C., Z.L., W.Z. and J.Y. ; software, S.C. and Z.L.