# Fourier Neural Operator Network for Fast Photoacoustic Wave Simulations
## 4 Discussion  
A key motivation for an FNO network is that a trained network can quickly produce accurate solutions for both forward and adjoint simulations. For a comparable simulation, the FNO network was \(\sim\)26\(\times\) faster on a 64 \(\times\) 64 grid and \(\sim\)15\(\times\) faster on a 128 \(\times\) 128 grid than k-Wave. This reduction in computation time with minimal loss in accuracy is ideal for applications requiring repeated evaluations of the photoacoustic operator. For example, advanced image reconstruction techniques, such as iterative methods, yield state-of-the-art results but are computationally expensive due to the repeated evaluation of the forward and adjoint operators. Therefore, the FNO network can greatly accelerate these methods by replacing traditional solvers for those operators. The FNO network is parameterized by the number of Fourier modes and channels. Increasing either parameter typically improves model performance but at the cost of increased memory and computation requirements. PDEs with generally smoother solutions require fewer modes to achieve a satisfactory solution. However, the photoacoustic operator contains broadband frequency information, which means that a higher number of modes is needed for an accurate solution. The optimal number of Fourier modes retained is related to the spatial resolution of the initial photoacoustic source and not necessarily the size of the computational grid, meaning that not all simulations with a computational grid of 64 \(\times\) 64 require the maximum 64 Fourier modes for the FNO network to produce an accurate solution. Hyperparameter optimization is especially important for simulations with large computational grids where limited GPU memory can become a problem. Optimizing the FNO for larger computational grids can be partially addressed via hyperparameter selection. There are other approaches that can be employed to further reduce GPU memory limitations. In this work, a 3D CNN architecture was used, but there are other CNN architectures, such as the recurrent 2D network, that are more memory-efficient [31]. Instead of solving for the full temporal solution in a single step, the FNO can be used iteratively to solve the wave equation for \(n\) time steps in each forward pass over the full time interval. Conventional solvers typically require sufficiently small time steps for solution stability, but the FNO is likely not limited by this requirement. Thus, it is possible that the FNO can use larger time steps and obtain an accurate solution for downstream image reconstruction and processing tasks. Figure 11: Zero-shot super resolution using an FNO trained on a 128 \(\times\) 128 \(\times\) 302 computational grid to perform simulations on a 224 \(\times\) 224 \(\times\) 302 grid. Images from the FNO and k-Wave simulations at t = 5 and zoomed regions are shown. (**Left**) Animal image of a lion from the testing dataset. (**Right**) Vasculature image to evaluate FNO generalizability. In this work, we chose to simulate the full spatio-temporal wave field to decouple the tasks of simulation from sampling. This allowed the same trained FNO to be used for any sensor configuration. The FNO could be configured and trained to directly output the sampled sensor data. This would reduce memory requirements but would also need to be retrained for each sensor geometry and configuration. A practical limitation in data-driven PDE solvers, such as the FNO network, is the need for high-quality training data. Traditional solvers are often used to create arbitrarily large datasets to train the network. Depending on the size of the computational grid, this can be computationally formidable, such as the case of 3D photoacoustic simulations. To create a large dataset in these scenarios, a high-performance computing environment would be needed to generate the training data in a reasonable timeframe. Transfer learning could also be employed to partially address this challenge after training the initial FNO model.