# Fourier Neural Operator Network for Fast Photoacoustic Wave Simulations
## 2 Materials and Methods
### Fourier Neural Operator Networks  
The FNO network [31] was adapted for solving the 2D photoacoustic wave equation. In the original implementation, the input \(a\) to the FNO was the first several time steps of the solution acquired using a conventional solver. Using insights from the physics of photoacoustic imaging, we determined that only the initial pressure distribution (time = 0) was needed as the input a to the FNO for solving the wave equation because the generated acoustic pulses propagate in an omni-directional manner. This removed the need for conventional solvers, and the full spatio-temporal solution could be obtained using only the FNO. To avoid instabilities and keep our solution bounded, we replaced previously used element-wise Gaussian normalizers with a peak-normalization scheme, where the initial source distribution map was normalized by its maximum value. We empirically found that the Gaussian normalizer was not appropriate for simulating wave propagation because the distribution of pressures over time at any given element in the computational grid was not normally distributed. The network begins by projecting the input \(a\) (initial pressure distribution) onto a higher dimensional latent representation using the fully connected layer \(FC_{1}\) with a single shallow layer (Figure 2). The dimensionality of this latent representation is defined by the hyperparameter termed channels. Four Fourier layers are then used to iteratively update the projected features. In each Fourier layer, the features initially undergo a Fourier transform, which plays a key role in enabling the network to efficiently learn mesh- and resolution-invariant features for solving the PDE. Features learned in the Fourier space are global by nature and represent patterns spanning the whole computational grid. In contrast, features learned in a standard CNN are local by nature and represent patterns spanning over a local region (e.g., edges and shapes). Figure 2: Neural network architecture for the FNO network. The input a (initial pressure distribution a) is mapped to a higher dimensional space using a fully connected layer (\(FC_{0}\)). The transformed feature is passed through four Fourier Layers (FLs). Finally, a fully connected layer (\(FC_{2}\)) is used to obtain the final output u (solution to the wave equation u) with the desired dimensions. The input goes through two paths in each Fourier layer. In the top path, the input undergoes a Fourier Transform FFT, linear transform R and inverse Fourier Transform iFFT. In the bottom path, the input undergoes a linear transform. Outputs from each path are summed together and undergo GeLU activation. The dimension of the feature representation for each operation is given in the parentheses. After the Fourier transform, the resulting Fourier modes can be truncated to optimize computational efficiency. This is useful as a regularization technique and for PDEs with smooth solutions that can be accurately represented with fewer Fourier modes, as previously demonstrated for the 2D Navier-Stokes equation [31]. Following the Fourier layers, the updated features are projected in \(FC_{2}\) to a higher dimensional representation with 128 channels in the hidden layer and finally to the desired dimensions in the final shallow layer to obtain the output \(u\) (solution to the wave equation). Through a combination of Fourier, linear and non-linear transformations, the FNO network can approximate highly complex and non-linear operators in PDEs. Channels and modes are the two main hyperparameters for the FNO network. Channels represent the dimensionality of the latent representation in the FNO network, and modes define the number of Fourier modes retained in each Fourier layer. Increasing the channel parameter generally increases the representational power of the model to learn more complex operators but can lead to issues of overfitting. There is no upper limit to the number of channels that can be used.