# Fourier Neural Operator Network for Fast Photoacoustic Wave Simulations
## 1 Introduction  
Photoacoustic imaging is a non-invasive hybrid imaging modality that combines the advantages of optical (e.g., high contrast and molecular specificity) and ultrasound (e.g., high penetration depth) imaging [1]. It has been applied for many preclinical and clinical imaging applications, such as small-animal whole-body imaging, breast and prostate cancer imaging and image-guided surgery [2; 3; 4; 5; 6]. Specifically, in breast cancer detection, tumors have been successfully revealed by single-breath-hold photoacoustic computed tomography (SBH-PACT) without the need of ionizing radiation and exogenous contrast agents based on the higher blood vessel density characteristics associated with tumors [7]. Multispectral photoacoustic imaging can be used for functional imaging, such as measuring blood oxygen saturation and metabolism in biological tissues [8]. In addition to applying multispectral photoacoustic imaging to differentiate oxym hemoglobin from deoxyhemoglobin in breast cancer, ultrasound and photoacoustic tomography (US-OT) can reveal differences in lipids and collagen in breast fibrolandular tissue, providing more clinically meaningful insights for diagnosis [9]. Photoacoustic imaging provides both structural and functional information that can potentially reveal novel insights into biological processes and disease pathologies [10]. In photoacoustic tomography (PAT), a tissue medium is illuminated using a short-pulsed laser. Optically absorbing molecules within the medium are excited and undergo thermoelastic expansion, resulting in the generation of photoacoustic waves that are subsequently measured using an array of acoustic sensors [1]. An image representing the initialpressure distribution can be reconstructed from the measured time-dependent signals using analytical solutions, numerical methods and model-based iterative methods [11; 12; 13; 14; 15]. A detailed understanding of parameters describing the imaging medium (e.g., optical, thermal and acoustic properties of the tissue) and the imaging system (e.g., arrangement and characteristics of the laser source and acoustic sensors) is needed to reconstruct a high-quality PAT image. PAT simulation is a highly useful tool that provides quantitative and qualitative insights into these parameters affecting image quality [16]. It is commonly used prior to experimentation and imaging to optimize the system configuration. It also plays an integral role in PAT image reconstruction and provides numerical phantom data for the development of advanced algorithms, such as iterative methods and deep learning methods [17; 18; 19; 20; 21; 22]. Simulating PAT image acquisition comprises two components: optical illumination and photoacoustic propagation. For this work, we are primarily focused on the photoacoustic propagation component. The equation for photoacoustic wave propagation can be solved numerically using classical methods, such as the time domain finite element method [23; 24]. These methods generally solve the equation via approximation on a mesh and can be computationally expensive, such as for large three-dimensional (3D) simulations. Recently, deep learning has been explored as an alternative method for solving partial differential equations (PDE) [25; 26]. It has the potential to greatly impact scientific disciplines and research by providing fast PDE solvers that approximate or enhance conventional ones. Applications requiring repeated evaluations of a PDE can greatly benefit from the reduced computation times offered by deep learning. Here, we provide a brief overview of three deep learning methods for solving PDEs--finite dimensional operators, neural finite element models and Fourier Neural Operators (FNO). Finite dimensional operators use a deep convolutional neural network (CNN) to solve the PDE on a finite Euclidean Space [27; 28]. This approach is mesh-dependent, meaning the CNN needs to be retrained for solving the PDE at different spatial resolutions and discretization.