Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that occurs when solving the Helmholtz equation. In this paper, a formulation is presented that addresses this difficulty. The problem of solving the two-dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method. According to this method, a trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfinite interpolation and the theory of R-functions. This ansatz is initially applied to the rectangular domain and later extended to the circular and elliptical domains. The acoustic field predicted from the proposed formulation is compared with that obtained from the two-dimensional finite element methods. Good agreement is observed in all three domains considered. Minor limitations associated with the proposed formulation and their remedies are also discussed.

翻译：物理信息神经网络为求解多个控制复杂物理现象的微分方程提供了另一种途径。然而，在求解亥姆霍兹方程时，梯度消失问题限制了其在预测声场方面的成功。本文提出了一种解决此困难的公式。通过试解方法，将求解具有给定边界条件的二维亥姆霍兹方程问题表述为一个无约束优化问题。根据该方法，利用超限插值技术和R-函数理论，构建了一个在训练过程之前即满足给定边界条件的试解神经网络。此方法首先应用于矩形域，随后扩展到圆形和椭圆形域。将所提公式预测的声场与二维有限元方法获得的结果进行了比较。在所有考虑的三种域中均观察到良好的一致性。文中还讨论了所提公式存在的微小局限性及其补救措施。