Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics

  1. Stefan Schoder

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Abstract

The generalization of Physics-Informed Neural Networks (PINNs) used to solve the inhomogeneous Helmholtz equation in a simplified three-dimensional room is investigated. PINNs are appealing since they can efficiently integrate a partial differential equation and experimental data by minimizing a loss function. However, a previous study experienced limitations in acoustics regarding the source term. A challenging but realistic excitation case is a confined (e.g., single-point) excitation area, yielding a smooth spatial wave field periodically with the wavelength. Compared to studies using smooth (unrealistic) sound excitation, the network’s generalization capabilities regarding a realistic sound excitation are addressed. Different methods like hyperparameter optimization, adaptive refinement, Fourier feature engineering, and locally adaptive activation functions with slope recovery are tested to tailor the PINN’s accuracy to an experimentally validated finite element analysis reference solution computed with openCFS. The hyperparameter study and optimization are conducted regarding the network depth and width, the learning rate, the used activation functions, and the deep learning backends (PyTorch 2.5.1, TensorFlow 2.18.0 1, TensorFlow 2.18.0 2, JAX 0.4.39). A modified (feature-engineered) PINN architecture was designed using input feature engineering to include the dispersion relation of the wave in the neural network. For smoothly (unrealistic) distributed sources, it was shown that the standard PINNs and the feature-engineered PINN converge to the analytic solution, with a relative error of 0.28% and 2×10−4%, respectively. The locally adaptive activation functions with the slope lead to a relative error of 0.086% with a source sharpness of s=1 m. Similar relative errors were obtained for the case s=0.2 m using adaptive refinement. The feature-engineered PINN significantly outperformed the results of previous studies regarding accuracy. Furthermore, the trainable parameters were reduced to a fraction by Bayesian hyperparameter optimization (around 5%), and likewise, the training time (around 3%) was reduced compared to the standard PINN formulation. By narrowing this excitation towards a single point, the convergence rate and minimum errors obtained of all presented network architectures increased. The feature-engineered architecture yielded a one order of magnitude lower accuracy of 0.20% compared to 0.019% of the standard PINN formulation with a source sharpness of s=1 m. It outperformed the finite element analysis and the standard PINN in terms time needed to obtain the solution, needing 15 min and 30 s on an AMD Ryzen 7 Pro 8840HS CPU (AMD, Santa Clara, CA, USA) for the FEM, compared to about 20 min (standard PINN) and just under a minute of the feature-engineered PINN, both trained on a Tesla T4 GPU (NVIDIA, Santa Clara, CA, USA).

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  1. Version published to 10.3390/app15020939
  2. PREreview

    This Zenodo record is a permanently preserved version of a PREreview. You can view the complete PREreview at https://prereview.org/reviews/14588713.

    This manuscript investigates the use of neural networks for the wave propagation problems. There are a few points that could be addressed.

    • Throughout the paper, the wavelength is kept constant as 1/2 and the domain length is 1. This is a constrained way of investigating the wave propagation as there are only two waves in one direction, as such the wave dispersion, pollution, reflection effects etc. can be monitored properly. Imagine that we have a music hall and a cello player plays a note A at 440 Hz. The wavelength will then be ~0.78 m and the room length would correspond to ~1.6 meters. If the first octave A note is played, then we would be in a room that is less than 1 m3 in volume. This is not a proper setting for the investigation of room acoustics problems. 

    • The manuscript places a large focus on the parameter s (sharpness); though, it is rather demonstrating in general that the model does not properly work, reporting error values around 98% - 99%! The only case that seems to have non-erratic results is that when the case approaches the classical Helmholtz problem for large values of s.   What is the benefit of publishing erroneous simulations? It would be more appropriate to omit the related sections where s is smaller than 1. 

    • The manuscript proposes to apply the Fourier feature extraction method in Section 4.5. However, equation (23) is just an alternative way of rewriting the analytical solution given by Eq. (5). The training data (X, y) pairs are generated in the current method according to the analytical solution or the FEM solution. For the case s = Inf, between the first layer and the output layer, the neural network is practically being trained for the value pairs p(x) = p(x). Would one practically need to perform a computational simulation for this case? 

    • The essence of neural networks (NN) and artificial intelligence is that one develops a model that is subsequently able to predict some unseen cases. In the current paper, can the trained NN be used for any other cases? For example, if the acoustic source is re-positioned to x=0.4, y=0.5, z=0.5, can we make reliable predictions? Alternatively, if the driving frequency (hence the wavenumber) is changed by a factor of 10 %, or the boundary condition on some part of the domain is modified, can the NN model deliver accurate results? 

    Competing interests

    The author declares that they have no competing interests.

  3. Version published to 10.20944/preprints202411.1848.v1