Vol.:(0123456789) 1 3 Iranian Journal of Science and Technology, Transactions of Civil Engineering https://doi.org/10.1007/s40996-020-00413-w RESEARCH PAPER Application of ANFIS–PSO Hybrid Algorithm for Predicting the Dimensions of the Downstream Scour Hole of Ski‑Jump Spillways M. Rashki Ghaleh Nou 1  · M. Zolghadr 2  · M. Shafai Bajestan 3  · H. M. Azamathulla 4 Received: 27 June 2019 / Accepted: 16 April 2020 © Shiraz University 2020 Abstract Ski-jump spillway (SJS) is one of the energy-dissipating structures, passing the food through the dams. In these structures, the downstream bed is scoured due to the collision of the high energy jet. It is essential to obtain information on the scour- hole dimensions to enhance the safety of the dam and related structures. The machine learning and data mining methods are efcient in predicting hydraulic phenomena. In this study, the geometric properties of the scour-hole including the hole expansion (width), maximum depth of the scour and its distance from the end edge of the spillway were predicted using a model by combining the adaptive neuro-fuzzy inference system (ANFIS) and the particle swarm optimization (PSO) algorithm. The results showed that the developed ANFIS–PSO model obtained maximum scour depth with a precision of R 2  = 0.94 and RMSE = 0.565, scour hole width with precision of R 2  = 0.95 and RMSE = 1.153 and its distance from the end edge of the SJS with an accuracy of R 2  = 0.96 and RMSE = 1.809. The evaluation and comparison of the simulation results indicated the superiority of the proposed model over other methods. Keywords Energy-dissipating structures · Machine learning algorithm · Scouring hole · Data mining · Optimization technique 1 Introduction Ski-jump spillways (ski-jump or flip bucket) are the energy-dissipating structures that greatly reduce the jet kinetic energy by throwing the jets into the air and impact to the plunge pool. These structures are one of the most common energy-dissipating structures. However, the main problem with them is that the jet issued from fip bucket still has the required energy for scouring downstream of the dam. If the collision area has diferent characteristics from those considered in the design, the scouring phenomenon may cause damages to the dam (Heller et al. 2005; Parsaie et al. 2016). The prediction of the scour hole dimensions downstream of the ski-jump spillways is one of the most important parts in design of the structure’s foundation. In general, the inves- tigation of the scouring phenomenon is not easy, due to the complexity of the comprehensive and simultaneous study of the water and sediment fux and the existence of several parameters afecting this phenomenon. Until now, several empirical relationships have been proposed to estimate the scour hole dimensions. Based on experimental data, Heng et al. (2013) evaluated 28 formulas to determine the maxi- mum scour depth downstream of the spillway. Most of the formulas were defned in a certain range of experimental data with a limited response interval. Hence, generaliza- tion of them is not reasonable for other conditions. Conse- quently, sometimes even for a particular structure, numer- ous formulas have been developed and with signifcantly diferent results. In addition, the construction of a physical model has many problems and limitations, and the exact efect of all parameters cannot be taken into account when determining the mapping between the parameters afecting * M. Rashki Ghaleh Nou mohammadrashki@pgs.usb.ac.ir 1 Hydraulic Structures, University of Sistan and Balouchestan, Zahedan, Iran 2 Department of Water Sciences Engineering, Jahrom University, Jahrom, Iran 3 Department of Hydraulic Structures, Shahid Chamran University of Ahvaz, Ahvaz, Iran 4 Department of Civil Engineering, University of the West Indies, Trinidad, Trinidad and Tobago