# Isoline Tracking in Particle-Based Fluids Using Level-Set Learning Representation
## 5 Results and Analysis
Additionally, the surfaces represented inside the fluid are often inferred below the particle positions, or sometimes surfaces disappear despite the presence of particles (see Figure 10a). On the other hand, the surface inferred via our method does not exhibit the problems seen with the 1-layer model's inferred surface and accurately represents the fluid's isolines. Figure 7: RMSE graph of the LSTM model. Figure 11 compares the surfaces inferred by the 1-layer model and our method. As seen in Figure 11a, the surface inferred via the 1-layer model restores the surface as if it were disconnected despite the particles being connected, and even biases the position of that surface upwards. In contrast, the surface inferred using our method is accurately and stably reconstructed near where the particles exist, without bias in any direction. Similarly,  
Figure 8: Visualization of the results from predicting fluid surfaces using (**a**) learning representation (1-layer) and (**b**) our method, respectively. Figure 10: Comparison-2 of visual differences between the learning representation (1-layer) and our method. Figure 9: Comparison-1 of visual differences between the learning representation (1-layer) and our method. in Figure 11b, the 1-layer model restores the surface of connected particles as if they were disconnected, showing a result biased downwards. On the other hand, our method stably restores the surface at the particle positions. Figures 12-17 show scenarios used to assess whether the trained model can be generalized and applied in various environments different from the training environment. Figure 12 shows the results of extracting the isolines of fluid fluctuating due to mouse interaction on flat fluid surfaces. As depicted in the figure, there was not a significant difference in quality between the original surfaces and the isolines tracked via our method. The proposed approach is efficient as it represents isolines through learning without going through the explicit level-set calculation process, and it not only accurately represents flat water but also stably captures the shapes of bubbles exhibited in splashing effects. Figure 13 depicts a dam breaking scene, where the fluid's isolines moving to the right due to the wave crest are well represented, showing no significant difference when compared with the original surfaces. Figure 14 presents a falling water scene, where the isolines in the splash caused by the strong impact due to gravity are acc  
Figure 11: Comparison-3 of visual differences between the learning representation (1-layer) and our method. Figure 12: Test scene (scene name: flat surface; red line: our method; black line: ground truth). Figure 13: Test scene (scene name: dam break; red line: our method; black line: ground truth). the results shown earlier. This paper demonstrated the restoration of complex fluid surfaces across various scenarios through experiments, consistently learning the representation of fluid surfaces in all results, not limited to specific scenes. As indicated by the test results in Figure 18, it was observed that values with high peaks in certain sections are generated due to the influence of gravity and external forces not included in the training data. The peaks that appear between 0 and 250 frames are caused by particles falling due to gravity and then moving in the opposite direction of their initial velocity due to the rebound effect when colliding with boundaries or other particles. The remaining peaks appeared when the user dragged with the mouse, demonstrating that the method can stably track the fluid isoline even when user interaction occurs. Figure 17 shows the result of the user interfering in the fluid simulation using mouse interaction. In this scene, our method also stably tracked the surface without any flickering or missing surfaces. In particle-based simulations, a level-set function is typically required to extract surfaces or isolines. Due to the absence of connectivity information inherent in conventional  
Figure 18: RMSE in each scene.