## 1 Introduction
### Problem Statement for Data-Driven Models in Hydrodynamic Responses  
Figure 1 indicates the framework of data-driven models, where the input \(\mathrm{U[k]}\) relates to the output \(\mathrm{Y[k]}\) through the intermediary system, marked as the trained model in the yellow box. The trained model is composed of two elements of deterministic \(\mathrm{X[k]}\) and stochastic \(\mathrm{Z[k]}\) components. The deterministic part defines what can be directly understood and relates between input and output through any mathematical models that are under the user's control, whereas the stochastic part lacks/evades any intuitive modeling and understanding. The stochastic component normally contains all residuals that cannot be accommodated in the deterministic component, comprising sensor noise, unmeasured disturbances, modeling errors, and unknown causes. In this regard, all elements in the stochastic components are lump summed as an independent identically distributed (i.i.d) random variable, or the specific case of which is white noise. In fact, all data-driven models endeavor to propose a trained model that can efficiently replicate the same output for the system excited by the inputs. However, when it comes to hydrodynamic responses, different challenges arise that can be summarized into the following:  
* Stochastic nature of environmental forces, \(\mathrm{U[k]}\). * Differentiation of deterministic and stochastic components in trained models. * Intrinsic limitations of surrogates. Considering this, the ideal is to train a network using a low-fidelity simulation surrogate that could reflect real-world data, i.e., training an ML model with analytical data and deploying the network directly on a vessel. However, it seems far-fetched since the simulation data lacks many features available in field data. An alternative method involves training the ML network directly with real-world data. However, as addressed, two problems are associated: First, lack of control over input does not let labeling the data accurately, or, in other words, the given label carries uncertainties. On the other hand, uncertainties in real-world data are often not explicitly identified and separated. Consequently, when the network is trained on such data, it learns from both the actual data and the accompanying noise/uncertainties. Therefore, the network may not be directly applicable to other scenarios where the uncertainty level varies; the combination of these two further complicates the problem. This fact somewhat undermines the scientific credibility of ML research solely based on the single data type in hydrodynamic responses and seakeeping. Despite control over inputs in model tests, experiments are expensive and sensitive to scale. CFD requires experiments for validation and is computationally intensive. The analytical synthesized data are easy to produce with a high level of controllability, even though they essentially lack the feature and entropy in higher data generation levels [7]. Figure 1: Underlying principle of data-driven models [9]. Figure 2a schematically illustrates the issue with the lack of entropy in the synthesized data. A new data value of 0.5 cannot be accurately attributed to either group. In contrast, the entropy in the high-fidelity data can positively influence the discernment of new data attribution. Therefore, using these techniques for ML ends up with over-optimistically and excessively neat, trained data that lack real-world applicability. Figure 2b perfectly pictures the relationship between different forms of surrogates for seakeeping. Addressing the stochastic nature of the seaway requires generating numerous scenarios, where high-fidelity data are essential but low-fidelity data are practical. Thus, the ideal approach involves utilizing simulation for computational efficiency and time-saving benefits while also incorporating/augmenting features from higher fidelity levels. This encapsulates the essence of the current study.