# Isoline Tracking in Particle-Based Fluids Using Level-Set Learning Representation
## 5 Results and Analysis  
Figure 6 shows the visualization after reconstructing the surface based on the level-set inference results using linear regression and LSTM models. Figure 6a illustrates that the surface reconstruction experienced artifacts due to the divergence of the level-set values predicted via linear regression. This divergence occurs because linear regression, typically used to learn linear structures of data, fails to learn the nonlinear data structure of the level set. Figure 6a shows the first frame of the simulation, and in subsequent frames, the predicted level-set values diverge, resulting in an inaccurate representation of the fluid surfaces (see Figure 6b). Figure 6b shows the visualization of the surface reconstruction based on the inference results from the LSTM model. Similar to linear regression, the figure shows the divergence  
Figure 6: Visualization of the results from predicting fluid surfaces using (**a**) linear regression and (**b**) LSTM models, respectively (red box: artifacts). of the predicted level-set values using LSTM. Unlike linear regression, it is visually apparent that the level set of the initial frames can somewhat predict the surface compared to later frames. The structural characteristic of LSTM, which ensures temporal continuity by using the level-set values of previous frames as input, leads to the accumulation of errors in the predicted level-set values. This accumulation results in an increasing error as the simulation progresses, which can be visually confirmed. Figure 7 shows a chart that calculates error values through RMSE for the level-set values inferred via the LSTM model for each frame. As previously mentioned, the error is small for the initial frames, but as the simulation progresses, the error values gradually increase, indicating a trend of growing discrepancies. Figure 8a shows the results of surface reconstruction using level-set values inferred via the learning representation (1-layer) model. Contrary to other results depicted in Figure 6, it can be visually confirmed that the surface has been somewhat tracked. However, there are artifacts where parts that should be connected are disjointed, or areas inside the fluid are incorrectly represented as the surface. Additionally, due to the lack of temporal continuity, flickering occurred when external forces not included in the training were applied during the simulation. The term \"flickering\" refers to the phenomenon where significant discrepancies in the reconstructed surfaces between consecutive frames lead to visually noticeable structural inconsistencies. Figure 8b shows the surface reconstruction using level-set values inferred from the model proposed in this paper. Visually improved results can be seen compared to the learning representation (1-layer) model shown in Figure 8a, but errors still exist in the surface reconstruction. It was also confirmed that the flickering phenomenon, which appeared in Figure 8a, occurs in the proposed model as well. This issue arises when the range of level-set values corresponding to the surface is lower than the allowed threshold. In traditional algorithms, surfaces above a certain threshold around a particle are directly represented during surface reconstruction. However, since the level-set values inferred from our method slightly differ from the ground truth, it is necessary to adjust the surface restoration using a threshold within an acceptable range. To address this issue, values had to be determined experimentally, and in this paper, 0.1 was added to the inferred \(\phi\) value. As a result, the problem was resolved, and issues where the interior of the fluid was mistakenly represented as the surface, as shown in Figure 9, were corrected, allowing for the creation of smooth surfaces. Figure 10 illustrates the differences in underestimation and overestimation of fluid isolines. In Figure 10a, the surface inferred via the 1-layer model tends to predict the surface isoline above the actual particle positions when there is significant particle movement.