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In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.

Offshore wind turbines are supported by various foundations, each with its considerations in design and construction. Gravity, monopile, and suction bucket foundations encounter geotechnical issues, while jacket and tripod foundations face fatigue problems. Considering this, a gravity foundation based on a steel skirt was developed, and a monopile foundation was analyzed for Pile-Soil Interaction using the p–y curve and 3D finite element method (3D FEM). In addition, for suction bucket foundations, the effects of lateral and vertical loads were analyzed using 3D FEM and centrifuge tests. Fatigue analysis for jacket and tripod foundations was conducted using a hotspot stress approach. Some hybrid foundations and shape optimization techniques that change the shape to complement the problems of each foundation described above were assessed. Hybrid foundations could increase lateral resistance compared to existing foundations because of the combined appendages, and optimization techniques could reduce costs by maximizing the efficiency of the structure or by reducing costs and weight. This paper presents the characteristics and research directions of the foundation through various studies on the foundation. In addition, the optimal design method is presented by explaining the problems of the foundation and suggesting ways to supplement them.