亥姆霍兹方程
可解释性
人工神经网络
偏微分方程
离散化
亥姆霍兹自由能
残余物
边值问题
应用数学
边界元法
计算机科学
有限元法
人工智能
算法
物理
数学
数学分析
量子力学
热力学
摘要
Discretization-based methods like the finite element method have proven to be effective for solving the Helmholtz equation in computational acoustics. However, it is very challenging to incorporate measured data into the model or infer model input parameters based on observed response data. Machine learning approaches have shown promising potential in data-driven modeling. In practical applications, purely supervised approaches suffer from poor generalization and physical interpretability. Physics-informed neural networks (PINNs) incorporate prior knowledge of the underlying partial differential equation by including the residual into the loss function of an artificial neural network. Training the neural network minimizes the residual of both the differential equation and the boundary conditions and learns a solution that satisfies the corresponding boundary value problem. In this contribution, PINNs are applied to solve the Helmholtz equation within a two-dimensional acoustic duct and mixed boundary conditions are considered. The results show that PINNs are able to solve the Helmholtz equation very accurately and provide a promising data-driven method for physics-based surrogate modeling.
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