# Isoline Tracking in Particle-Based Fluids Using Level-Set Learning Representation
## 2 Related Work  
### Surface Reconstruction of Particle-Based Fluids  
Originally developed to solve problems in astrophysics, the SPH technique [7], a prominent particle-based fluid simulation method, became popular in fluid simulations due to its ability to assign physical properties to particles instead of a mesh. This led to research into various methods of solving partial differential equations using SPH [10; 11; 12; 13]. The instability of fluid simulations and their significant resource consumption make the search for methods to achieve stable simulations within domains [14; 15; 16; 17] and natural representation of particle-based fluids at low cost [18; 19; 20] an ongoing popular topic. In typical fluid simulations, techniques are used to realistically represent the fluid surfaces based on the material properties of certain particles or to reflect different characteristics like honey or water in the physical properties affecting particle movement [2; 13; 21]. Extensive research focuses on reconstructing meshes from point cloud structures to reduce resource usage and computation time while visually representing fluids naturally. These studies are widely referenced in related research fields.  
To represent the surface of a fluid, it is necessary first to identify surface particles among all fluid particles. In grid-based simulation frameworks, methods like the level-set method [22], particle level set [23; 24; 25], and VOF (Volume of Fluid) [26] have been proposed. In the meshless (Lagrangian) simulation framework used in this paper, [PERSON] [27] introduced the blobby sphere approach. This method extracts surfaces from a scalar field composed of the sum of radial functions centered on each particle. However, it faced issues of surface irregularities when particle density was high or low. [PERSON] and [PERSON] [28] obtained smoother surfaces by computing scalar values from basis functions, which are the weighted sum of neighboring particles' weights and radii, and then sampling this scalar distance function on a grid followed by a smoothing pass. Later, [PERSON] [29] improved this method proposed by [PERSON] and [PERSON] by tracking the particle-surface distance every frame. This approach adjusted the distance of each particle to the surface and propagated the distance information from surface particles to internal particles, thereby realigning the particles. This method also enables smooth surface generation for both fixed-radius particles and adaptive particle size in fluid simulation techniques [20] that increase memory efficiency and reduce computational costs.  
MC [5] was proposed to efficiently construct high-resolution 3D surfaces. When using MC to generate surfaces, the volume of an object is divided into small cubic voxels. The algorithm constructs triangles based on a pre-allocated MC table in memory, approximating the surface within each voxel. These triangles are created by interpolating vertices on the edges of voxels based on the data values at their endpoints, and connecting the results in a smooth, continuous surface across the entire volume. While individual voxel computations allow for easy parallel processing, complex topologies may lose features like sharp edges. Also, additional information for each voxel, including vertex position, normals, and connectivity data, must be stored, increasing the required memory amount for high-resolution grids and large-scale data.  
DC [6] addresses the loss of sharp edge features in MC by effectively handling complex topologies through the integration of data value gradients. It does not solely rely on predefined tables but iteratively evaluates data points within each voxel, calculates gradients, and determines intersections where the surface crosses voxel edges. Utilizing these calculated intersections can enhance accuracy, but it also increases computational complexity. Efficient implementation requires additional optimization techniques, and there are some constraints to consider.