# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 6 Conclusions  
This study demonstrates the feasibility of using PINNs to model 3D sound fields in a room with sound-hard walls governed by the inhomogeneous Helmholtz equation. The prediction accuracy was assessed by a validated reference FEM simulation in openCFS, which took, on average, 15 min and 30 s to compute on a standard notebook CPU. Three different network architectures and feature engineering were used to improve the pressure field prediction. The most significant improvement was reached by ingraining the characteristics of the Green's function based on modal expansion via feature engineering as additional input to the PINN. An explorative hyperparameter study provides intuition about the behavior of the standard PINN concerning prediction accuracy. The network depth and width, activation functions, training routine, fine-tuning with different optimizers, backends, and learning rate adjustment improved the prediction capability for a simple, fully connected neural network architecture from an error of 0.45% to 0.019% (\(s=1\) m). The backend TensorFlow 1 on the Tesla T4 GPU with Cuda 12.2 and Python 3.10 had the best performance. The layer-wise local adaptive activation function network improved the prediction accuracy of the tested case. However, the neuron-wise algorithm, introducing more additional degrees of freedom, might further improve the capabilities. Residual-based adaptive refinement improved the prediction accuracy. The refinement workflow connected both structured and randomly unstructured refinement. In each stage, 150 data points were added, adding about 1.5% additional data points. The convergence to the accurate solution reached relative errors as low as 0.28% for distributed sources from the initial 0.35%. In the case of \(s=1\) m, the relative error improved by 60%, using adaptive  
Figure 10: Pressure results normalized by the maximum pressure of the reference (FEM, analytic) depending on the \(x\) coordinate at a line with \(y=z=0.5\) m inside the domain. refinement. In the case of \(s=0.2\) m, the relative error was reduced drastically by adaptive refinement being similar than the error computed by the mesh study of the FEM results. Using random Fourier features in the multi-scale Fourier feature network did not yield a higher accuracy. Studying the transformation behavior of the input by random Fourier features revealed visibly that the transformation does not reduce the complexity of the prediction. These visualizations led to the idea that using dispersion relation-based Fourier features may significantly enhance the prediction capabilities and the generalization of the neural network architecture concerning alternative source distributions. This feature-engineered architecture substantially reduces the number of trainable parameters that reach 5% and the training time on the GPU to less than a minute. This fast prediction outperforms the FEM analysis. Also, the accuracy is improved by three orders of magnitude for \(s\rightarrow\infty\), (10\({}^{-3}\)%). However, this is still two orders of magnitude less accurate than the pressure result of the FEM solution (10\({}^{-5}\)%). This work underscores the potential of PINNs as memory-efficient alternatives to traditional numerical methods for low-frequency room acoustic simulations. For the case of \(s=1\) m, the errors were reduced from initially 0.20% to 0.19% by a Gaussian process-based Bayesian hyperparameter optimization. This article presented significant progress regarding the generalization of PINNs for confined sources. Challenges regarding the explainability of the network learning capabilities [53] and the generalization to multi-source and different room geometries remain. These challenges will be subject to a follow-up study. Transitional behavior for very localized sources will be a focus of upcoming work, including explainability. This scientific work was significantly supported by the COMET project ECHODA (Energy Efficient Cooling and Heating of Domestic Appliances).