# Fourier Neural Operator Network for Fast Photoacoustic Wave Simulations
## 5 Conclusions  
Solving the 2D photoacoustic wave equation with traditional methods typically requires a fine discretization of the computational grid and can be time-consuming to complete. Deep learning methods directly learn from data to solve PDEs and can be orders of magnitude faster with minimal losses in accuracy. In this work, we applied the FNO network as a fast data-driven PDE solver for the 2D photoacoustic wave equation in a homogeneous medium. The photoacoustic simulations generated by a traditional solver with the k-Wave toolbox and the FNO network for the breast vasculature testing dataset were remarkably similar, both visually and quantitatively. The RMSE between the k-Wave and FNO simulations was several orders of magnitude smaller than the pressure intensities. Model generalizability is a highly desirable property for an FNO network. If the trained FNO network is to be a reliable alternative to traditional PDE solvers, then it needs to be capable of solving the photoacoustic operator for any arbitrary initial pressure source within an acceptable degree of error. Having a generalizable network minimizes the need to retrain the network for instances not observed in the training dataset. The generalizability of the trained FNO network was evaluated using phantoms not in the training data. Shepp-Logan, synthetic vasculature, tumor and Mason-M phantoms were used for 64 \(\times\) 64 simulations, and the vasculature phantom was used for the 128 \(\times\) 128 simulation. In general, the FNO network and k-Wave simulations were visually similar. The RMSE was relatively small for these phantoms but was larger than those in the testing dataset. This indicated that the FNO was overfitting the training data. These results provide evidence for the FNO network being generalizable and can be used for simulations with any arbitrary initial pressure source. Moreover, the FNO network was learning the photoacoustic operator and not memorizing specific solutions related to the training data. In this work, the FNO network was trained for solving the 2D acoustic wave equation in a homogeneous medium. Simulations with homogeneous media are widely used in many applications because the spatial distribution of heterogeneities is often unknown. Nevertheless, an FNO network can be used for simulations with heterogeneous media by providing the spatial distribution of medium properties as additional inputs to the FNO network. During the training process, the FNO network can leverage these inputs and learn from training examples generated with varying heterogeneous media to perform simulations with heterogeneous media. The operator for a heterogeneous medium is more complex than that of the homogeneous case. Thus, a larger and more diverse dataset is likely needed to adequately train the FNO network. The significance of our work is that it provides a comprehensive study on solving the 2D photoacoustic wave equation with an FNO, demonstrating its application potential in downstream reconstruction and image-processing tasks. The FNO performs orders of magnitude faster than traditional solvers and can generalize to unseen out-of-domain datasets. However, for simulations in a larger computational grid, a higher Fourier mode is necessarily required to learn the broadband frequency residing in the whole time series data. Consequently, it increases GPU memory and can be an intractable problem while simulating in a larger domain, which usually requires the output of more time steps. Furthermore, the current framework is limited to simulating the 2D photoacoustic wave equation. Simulating the 3D photoacoustic source to output entire 4D time series data is impossible to handle with modern GPUs. In addition, zero-shot super resolution using an FNO tends to output blurry images due to the insufficient number of Fourier modes used to train the network to support broadband frequency components in a larger computational grid. In the future, research addressing these issues will benefit the use of FNOs in more complex settings.