外文翻译基于神经网络的桩群冲刷评估的应用中英文

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1、附录1基于神经网络的桩群冲刷评估的应用土木工程系，印度理工学院，孟买邦，溥维区，孟买城400 076印度认可 2003年6月2日摘要：有桩建筑与海洋环境间的相互作用是相当复杂的。不管多么有实力的实验室,作为原始的研究评估冲刷程度，冲刷的几何形状到通用的、正确的都是很困难的。其中的一个不确定的原因是统计曲线的匹配技术的局限性，就是通常用来分析搜集到的数据的技术。因此，当前的工作是努力实现其他的数据挖掘技术神经网络分析来处理冲刷数据的分析。神经网络能够绘制一个随机的输入向量具有随机的输出向量在一个自由的模型里，不象原始的非线性变化方法。不同的网络模型被开发出来，用来预测冲刷坑的深度和宽度，这个是一

2、群支撑码头的桩子，该码头位于日本的一个海岸，通过浪高，浪的周期，水深和桩的直径等输入参数，也有桩的Reynolds值，最大浪的质点运动速度，最大浪的切速度，保护物变量，和KeuleganCarpenter值。该网络依靠循环训练得到发展，通过回溯和迭代算法。测试结果显示，神经网络方法能提供一个更好的统计匹配处理办法。个别的输入参数比组合的输入参数产生更好的结果。冲刷的深度是可以预测的更准确比冲刷的宽度。一个权值矩阵列出了在任何给定的位置的使用。关键词：神经网络; 桩冲刷; 群桩; 冲刷数据分析; 群桩冲刷1.引言在海洋里安置的许多结构必须安装桩来支撑，因为海底的土壤是松散的，不稳固的。大部分的桩

3、被放置在受侵蚀的海底，因此估算桩周围的冲刷是设计时的要点。冲刷能够引起建筑基础的损坏，导致整个建筑的垮塌。海床上的沉积物将被清除，如果它的重量不能够抵消流动的力类似摩擦力。当振荡波和水流被阻挡通过一个垂直的桩，就会带走桩周围的沉积物，在桩的周围形成一个倒圆锥的洞。越来越多的沉积物被移走，洞长大了，悬浮的沉积物被水流带到了凹陷的地方。时间的推移，形成的坑变的深远。达到最大的冲刷深度，准确的定位在各种问题，关于冲刷程度的评估。当两个或更多的桩在空间上很近的话，之间的距离超过3倍的直径的时候，形成的洞合并到一起，最终形成全部的侵蚀，一个倒圆锥的形状。水力冲刷垂直的圆柱体产生一个过来的研究话题。然而，

4、最新的研究针对在诸如桥梁码头对应的单方向的河流的建筑物。研究垂直桩服从摆动波的冲刷运动是最近相关的内容。除了大部分的调查研究的实验。2.网络发展神经网络由相互连接的神经元组成。每个神经元或者节点是一个独立计算单位（图 1），使用以下方程式进行计算： 在这里，是输出的神经元；; 是输入值；; ; 是权值；是偏差值；是计算函数，代表的函数表达式如下：非典型的神经网络用在现场研究显示在图2。被称做FF类型的网络，只是正方向进行估算。神经元分为三层，即输入，隐藏和输出层。神经元的输出结果构成了网络的输出结果。链形线路拥有向后的连接有时提供好的知识。一个例子链形线路显示在图3，输出的形式神经元能够反作用

5、到隐藏神经元通过前后关系。连接权值最初是选择的随机数，通过一段实践后再确定下来。选择性的实践过程需要用到，由于两个不同的计划，即BP和CC，目的是训练FF网络，来适应链形线路。任何练习运算法则的目的是最小化全局的错误；定义如下：此处的和是网络和目标输出到如何第个输出节点。总和计算完成遍历所有节点，遍历所有练习模式。一对输入，输出值建立一个训练模式。后传播运算法则计算错误作为平均值，分配它向后从输出节点到隐藏节点然后到输入节点。采用梯度下降原理在权值中的改变的做法是关注于错误梯度的负面。在此 是权值在任何两个节点之间；，是改变在第个权值和的反复；是动力要素和它的知识比率。知识比率管理权值变化的大

6、小，作为每个结果在全部的错误里。动力要素是防止权值在反复训练时的振动，并且加速在平的错误表面上训练。此时研究这些值被选择的不同值直到集中到达，当更深的反复或者训练循环开始，没有导致全部错误的对比值。 瀑布关系训练法则时定位在展开一个最适宜的网络体系和它开始没有任何隐藏的节点。它遵循以下步骤完成训练。1）只用输入和输出。2）训练网络遍历训练数据通过设置梯度规则。3）添加一个新的隐藏节点。连接它到所有输入节点就像连接到其他已经存在的隐藏节点一样。训练这些节点，基于最大化的全部关系S给出在此，是样式的隐藏节点的输出；是所有样式的平均输出；是网络输出错误对于输出节点在样式下；是平均网络错误遍历全部的样

7、式。通过一个个的训练数据，调整新输入节点的权值在每次训练之后，设置直到S不能稍微的改变。4）一旦完成新节点的训练，该节点被安装作为一个隐藏的网络节点。输入端权值被固定，和输出端权值不再次训练。5）到步骤3），并且重复该过程直到网络达到一个最小错误在一个给定的反复训练值中。反向算法可以被修改到Jordan-Elman scheme目的是训练正确的网络在以下的方法中。(1)初始化前后关系单位(2)执行以下步骤对于每一个训练样式； A）输入一个样式并向前传播它通过这个网络。B）计算误差通过比较计算输出和目标输出值。C)向后传播这个误差值。D）计算权值改变。只做在线训练权值。E）计算新的状态在上下文单

8、位依照引入的。(3)只做离线训练，采集权值神经网络工作细节如同不同的训练法则的描述，可以在6和7中显示。执行发展的网络需要以下帮助，1）绘制一个估计目标值的离散表2）评估相关系数，给出在此是网络输出，是目标输出，是测试样式数量。(4)决定效率的系数，给出3.数据Bayram 和 Larsen写的报告关于冲刷桩的被用做一个很好的测试。这是因为他们详尽的观察和相称的直接原始观察在大量的基于实验的研究报告。数据聚集在一个200米长调查码头，在Ajigaura Beach日本的太平洋海岸上。码头被设计在由海岸到海里，并且被9群立柱，每群有4根立柱0.6米直径在2.67米在图4中显示。每一个桩群彼此之间

9、距离是30米。一个群的冲刷洞大小是可用的环境参数通过56周测量从1984年8月到1989年9月。这立柱群在码头靠海的一端，远离断浪的影响，流行的摆动的浪。冲刷的尺寸与平衡状态相符，和浪高，浪的周期属于在前描述的周期平均值。沉积物的平均颗粒大小是0.2毫米。即使在桩群的中心到中心的距离比较大（在4.5倍的个体桩的直径），桩群有一个共同的冲刷洞，深度，高度在图表4（b）。通过研究冲刷流相关评估，冲刷流尺寸用几个可靠的不相关的群的个体参。他们是Reynolds数，Re；Keulegan数和 Carpenter数，遮蔽参数，；和Sediment值，。关于这些数的表达式如下： 在此 是最大水质点速度是浪

10、的运动（取决于线状浪的理论），是桩的直径，是海水密度，是浪的周期，是相对于静止海床的速度，定义：是浪的摩擦因子，是沉积物比重；是重力加速度，是沉积物的二次平均直径。在此次研究中用的不同参数列表在表1中给出。通过早期的研究组成的这些参数的参数群来评估冲刷洞的尺寸，一些在神经网络的调查工作已经提前作出报告，使用特定的参数代替他们全部的参数产生更好的结果，因为增强了灵活性在那种方法适应完成。因此确定开发一个网络拥有输入变量，变量名是，；水深，；有时不同在一个给出的位置，也做；是保持确定的含义。无论以后注意到后来的恒量不匹配到成果，只用输入一个参数来代替：结果是所有的4个恒量可能包含在权值里，网络存在

11、偏差。这可能导致在消除误差的网络训练里。更深地，冲刷流的尺寸一般不依赖沉积物的属性。网络由4个输入参数组成，名字是; ; 和，是第一个可以预报的冲刷深度，:作为每一个标准的实践，在训练中使用的80%的数据，而保留了20的雇佣测试。可以使用的SNNS软件是使用软件的发展的。4.冲刷深度预测冲刷深度的离散图，依靠不受约束的因数；; 和分别显示在表里，57。分布状态清晰的指出了非常无规律的，非线性变化的跟随; 和;证明了使用神经网络随机映射图技术目的是评估内在的关系。可以得出相同的结论在离散图，依靠; 和变量，分别显示在表810中提到关于冲刷深度预测的输入节点是依靠浪高，浪的周期，水深和桩的直径的。

12、输出节点与估计的值适应的：几个拓扑实验被研究，目的是在准确的结果中找到最好结果。在前面的段落里，所有的相关实验使用了两种网络和三种训练方法。在每个过程里，选择隐藏节点，学习率，因数和循环次数通过实验给出，一直计算，直到没有增长的误差被发现为止。训练方法和测试样例的选择包括被随机选的，直到最好的结果出现。这里用到了46种训练样例和13种测试样例。训练用例不包括在网络测试中。最正确的结果关于Feed Forward Back Propagation：4-6-6-1显示在图2里。比较结果网络产生的值和目标值在图例已经给出了。因此，可以在使用四个因子的得到神经网络能够捕获值；能够反映出一个高的相关系数

13、值0.99，低的平均误差值2.6%，高的令人满意的效率值0.98。训练权值矩阵在表2中给出。该表可以应用该网络到其他的使用通用回归方法的领域。附表A，给出了数学表达式，目的是使用该矩阵的帮助，得到冲刷深度的输出值。 表3给出了不同网络（随后的）的性能，这些网络是在处理不同的输入输出之间的研究使用的。它给出了最好的算法，网络拓扑，达到误差目标的循环数，学习率和因子值，还有，等。传统的输入输出参数的关系是通过多项式表述的，因此，通过有价值的探索怎样神经网络评估值对比统计的方法。内部的独立参数; 和变量的变化率，冲刷深度是因变量。因为桩的直径在现有资料里是一个常数，所以没有作为一个自变量使用。多项式

14、如下：在此，是因变量，; ; 是自变量，; ; ; 是常数是通过二次方程式得到的。据此归纳出来。在此，是冲刷深度；是浪高；是浪周期；是水深； 上面等式的正确性在训练数据得到的，经过相关的测试数据测试验证的。这显示在一个表名值变化率的离散图表里，图12。表下注明是0.91， 是6%，效率系数是0.75，标志着相关的适用程度。一个定性的对照关于神经网络模型的表达式通过图11，图12，如同定性的对照关于;；等参数的，由此显示出该论文中提及的神经网络是比传统的表达式正确的。根本的原因是网络是一个自由模型途径，因此具有非常大的自由度，不象表达式有局限性。调查照例叙述冲刷深度对于必然的几个描述流体不同方面

15、的参数。这包括; , 10。电子网络（详见表3）发展了从,和值中得到的方法。图1315给出的测试结果，分别连同相应的; 和值。图表显示了较差的表现能力，关于这些网络对比于图11。图11里值由; ; 值得到。在这3个图里。图1315中的值是从值得到的，合理的有建设性的依靠的值对比和变量：这在Bayram 和 Larsen方法里也得到了同样的数据。因此与的线性表达式如下；当这种关系被用关于测试数据设置测试后，产生一个对照表，如图16，该图显示等式16式不令人满意的。作为依靠产生的神经网络的值在当前研究里的是相当准确的。为了查看是否是使用全部的变量; , ;和加在一起提供一个更好的预测值；另一个神经

16、网络因此发展起来。实验拓扑开始尝试使用。图17显示最好的测试结果在所有的实验之外根据离散图表。图下注释非常低，是0.71，是20.73，然而很高，是10.7%。如果我们比较这个网络基于所有参数，我们较早开发的一个使用变量; ; 和；我们发现新做的这个摇好的多，也是我们期望的。因此发现使用所有参数没有导致任何好处，对比于独立变量; ; 和的早期讨论。这表示出网络训练的灵活性付出的代价。5. 冲刷宽度预测与一个桩群周围的冲刷洞的深度一样，它的宽度是设计者们普遍需要了解得另一个参数。接下来的研究涉及无范围形式的冲刷洞的宽度的预测，例如，;应用; ; 和；为此目的发展起来的神经网络有一个输出点和四个输

17、入点。像前面的冲刷深度网络类型的几个实验中，训练计划在调试网络的过程中形成。这仅仅是在相关系数不超过0.52时取得了有限的成功，并且不低于12.7%（这与拓扑图a4-10-1相对应，并且JordanElman算法的0：0001，见表3）图18显示了相对于价值目标的基本的网络产出。能够看出来与前面冲刷深度预测的例子不同，这里的冲刷洞的宽度不是最终依靠; ; 和因素。 为了预测令人满意的洞的的宽度可能需要另一个不可计量的因素像土壤特性和与其它码头临近。看下图810，显示了; 对于的变化程度，并且也暗示了这些变量之间的模糊的联系。Bayram 和Larson 5也在用统计曲线匹配估计中有相似的失败。

18、它显示了冲刷洞的几何形状很难用任何一种数学模型来计算。 做过尝试来验证通过联合网络来更好地预测冲刷宽度（也可以是冲刷深度）是否可行，这种联合网络有四个输入点（; ; 和）和两个输出点（和）。应用了几个拓扑实验。这在给出估计价值（同时包括价值）的可以接受的准确程度方面也失败了。在这种情况下被调试的网络（涉及10，000个循环）有拓扑4-9-9-2, 0:005;这导致了0:76和 0.53同时 13.6%并且和13.6%. (见表3)。最后值得提到的是估计冲刷范围的更完全的网络应该涉及到许多其他的输入参数，这些参数控制着冲刷得进程，例如，柱的直径，沉淀物尺寸，水流，河床的形态特征，浪的方向，随意

19、性，柱的间隔和他们的方位（见Ref. 10），但是，这需要意味着巨大成本支出的广泛的数据收集程序。6.结论前述章节提供了关于太平洋日本海岸的调查码头的立柱群的冲刷数据的分析介绍。1. 发现预报一个冲刷浪的程度，对于在海洋里的立柱，是可以改良的，如果神经网络被作为一个工具，在适当的位置处理矿业数据，通过传统的技术统计曲线相匹配。复杂的神经网络被比喻为衰退分析，因此被用来解决冲刷浪评估是必要的。表2中给出的权值矩阵连同图12一起可以用来预测无量纲的冲刷浪的程度，从输入的浪高，浪的周期，水深和立柱直径。2. 神经网络数据模型的灵活性在输入参数是排列组合的时候发现是痛苦的，胜于个别的，或者是毒药一个冲

20、刷浪参数，通过一个简单的网络预测。3. 预测是比较困难的，甚至是使用了神经网络，同样是有价值的，通过考虑其他输入因子，比如当地情况，土壤特性和其他相近立柱之间的关系。4. 网络训练可以改良通过网络类型的适当选择（顺序的，或循环的），训练算法（回溯法或迭代法或JordanElman法）和网络内部的控制参数（学习率，因子）附录A.该表达式得到冲刷深度1.输入节点权值求和：在此是第个隐藏节点的和，是输入节点的总数，是输入节点的和隐藏节点的关系，是第个输入节点的变量。2.转换输入权值：在此是第个隐藏节点的输出值。3.隐藏节点求和：在此，是第个节点的总和，是隐藏节点的总数，是第个隐藏节点和第个输出节点的

21、关系。4.转换总和权值：在此是第个输出节点的输出值。附录2Estimation of pile group scour using neural networksA.R. Kambekar, M.C. Deo*Department of Civil Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, IndiaAccepted 2 June 2003AbstractThe interaction between ocean environment and pile structure is so

22、 complex that despite considerable laboratory as well as prototype studies estimation of scour depth and its geometry in a generalized and accurate form are still difficult to make. One of the reasons underlying this uncertainty could be the limitation of the statistical curve fitting technique, com

23、monly employed to analyze the collected data. The present work therefore attempts to carry out scour data analysis using another technique of data mining: neural networks. Neural networks have ability to map a random input vector with the random output vector in a model-free manner unlike the model

24、oriented non-linear regression methods. Different networks were developed to predict the scour depth as well as scour width for a group of piles supporting a pier situated at a coastal location off Japan using the input of wave height, wave period, water depth and pile diameter as well as pile Reyno

25、lds number, maximum wave particle velocity, maximum shear velocity, Shields parameter and KeuleganCarpenter number. The networks were of feed forward as well as recurrent type trained using back propagation and cascade correlation algorithms. The testing results showed that the neural network could

26、provide a better alternative to the statistical curve fitting. Individual input parameters yielded better results than their grouped combinations. The depth of scour was predicted more accurately than its width. A matrix of weights is specified for use at any given location. 2003 Elsevier Ltd. All r

27、ights reserved.Keywords: Neural networks; Pile scour; Pile groups; Scour data analysis; Pile group scour1. IntroductionMany structures installed in the ocean are required to be supported by piles because of the weak and unconsolidated soil conditions commonly encountered below the sea. A majority of

28、 the piles are located on eroding bottoms and hence estimation of scour around the piles is essential in design. Scouring can cause failure of the structure toe, giving rise to collapse of the entire structure.A sediment particle at the sea-bed is removed from its position if its weight is unable to

29、 counteract the dislodging forces of fluid drag and lift as well as the skin friction. When oscillating waves and currents are obstructed by a vertical pile complex turbulent flow accompanied by vortex formations around the pile give rise to the removal of sediment such that a hole resembling that o

30、f an inverted cone starts forming around the pile. As more and more sediments are removed and the hole grows new suspended sediments are also brought in by the flow and dropped into the pit. With passage of time a state of equilibrium is reached and further scouring is inhibited. This state correspo

31、nds to the maximum scouring depth, which is aimed at in any problem of scour depth estimation. When two or more piles are closely spaced with their spacing greater than about three times the individual diameters, the holes around the piles merge culminating into a global scour having the same shape

32、as that of the inverted cone 1.Hydraulic scour around vertical cylindrical objects had been an old topic of study. However earlier studies were aimed only at structures like bridge piers facing the unidirectional river flows. Investigation of scouring action on vertical piles subjected to oscillator

33、y waves is relatively recent. Out of these works most pertain to laboratory based investigations. Experiments in labs attempted to relate scour depth and its geometry to flow and sediment characteristics expressed in terms of dimensionless parameters like wave steepness, and relative water depth 2,

34、Reynolds number, KeuleganCarpenter number, Shields parameter and sediment number 3. While laboratory works carried out under controlled conditions provided a useful insight into the mechanism of oceanic scour, applicability of these results to the field has always remained doubtful. This is due to t

35、he failure to simulate real sea conditions like random waves, multi-directionality of waves and interaction with currents as well as the difficulties in converting laboratory results to prototype.Field studies oriented towards scour estimation are therefore highly desirable. But they are very few an

36、d almost non-existing in respect of vertical piles. Exceptions to this include Palmer 4 and Bay ram and Larsen 5. The latter study pertains to scour depth estimation for groups of piles supporting a pier off Japanese coast. The present investigation uses the same data-base to estimate scour but empl

37、oys a different tool of analysis: neural networks. Its motivation stems from the observation that all past works have used the technique of regressionlinear as well as non-linearto analyze and synthesize the collected data. Regression techniques are model-oriented and hence rigid in that they first

38、assume some kind of fixed relationship between the input and the output beforehand. On the contrary neural networks are data-oriented and flexible and come up with their own dependency structure in between the input and the output. It was therefore thought worthy to examine if substitution of the co

39、nventional statistical curve fitting by the network, results in obtaining more accurate estimation of scour in the field.2. Development of the networksA neural network basically consists of interconnected neurons. Each neuron or node is an independent computational unit (Fig. 1), which works as per

40、the following equation, where, y is output from neuron; x1; x2; x3;are input values; w1;w2;w3; areconnection weights; is bias or threshold value; f is transfer function, typically sigmoid function given by,Atypical neural network used in the present study is shown in Fig. 2. This is called Feed Forw

41、ard (FF) type of network where computations proceed along the forward direction only. There are three layers of neurons, namely input, hidden and output layer. The output obtained from the output neurons constitutes the network output.Recurrent networks with backward connections some times provide g

42、ood learning. An example of such a recurrent network is shown in Fig. 3, where the output form the output neuron can be fed back to the hidden neurons through a context unit. The connection weights are initially chosen as random numbers and then fixed by following a training process. Alternative tra

43、ining processes are available, out of which the present study utilized two popular schemes, namely Back-propagation (BP) and Cascade Correlation (CC) in order to train the FF network and JordanElman algorithm to cater to the recurrent network.The goal of any training algorithm is to minimize the glo

44、bal (average sum squared) error E; defined below:where on and are network and target output for any nth output node. The summation has to be carried out over all output nodes and over all training patterns. A pair of input and output values constitutes a training pattern.The Back-propagation algorit

45、hm calculates the error E as per Eq. (3) and distributes it backward from the output to hidden and then to the input nodes. This is done using the steepest gradient descent principle where the change in weight is directed towards negative of the error gradient, i.e.Where w is weight between any two

46、nodes; are changes in this weight at nth and en 2 1Tth iteration; is momentum factor and his learning rate.The learning rate hgoverns the size of the weight change as per the effect of the weight on the total error. The momentum factor a prevents weight oscillations during training iterations and al

47、so accelerates the training on flat error surfaces. In the current study these values were selected by varying them till convergence was reached, i.e. when further iteration or training cycles did not result in reduced value of the total error.The Cascade Correlation training algorithm is aimed at e

48、volving an optimum network architecture and it starts without any hidden nodes. It follows following steps to complete the training.(1) Start with inputs and outputs only. (2) Train the network over the training data set by the gradient rule. (3) Add a new hidden node. Connect it to all input nodes

49、as well as to other existing hidden nodes. Training of this node is based on maximization of overall correlationS given by:Where Vp is output of the new hidden node for pattern p; V is average output over all patterns; is network output error for output node o on pattern p; and is average network er

50、ror over all patterns. Pass the training data set one by one and adjust input weights of the new node after each training set until S does not change appreciably. (4) Once training of the new node is done, that node is installed as a hidden node of the network. The input side weights are frozen, and

51、 the output side weights are trained again. (5) Go to step (3), and repeat the procedure until the network attains a prespecified minimum error within a fixed number of training iterations.The Back-propagation algorithm can be modified into the Jordan-Elman scheme in order to train the recurrent net

52、work in the following way.(1) Initialize the context units, (2) Execute the following steps for each training pattern: (a) Input the pattern and propagate it forward through the network. (b) Calculatethe error by comparing the computed output and the target output. (c) Back propagate the error. (d)

53、Calculate the weight changes. (d)Doonly on-line training, i.e. weight adoption. (e) Calculate the new state of the context units according to the incominglinks. (3)Doonly off-line training, weightadoption.Details of working of neural networks as well as description of various training algorithms can

54、 be seen in Refs. 6,7 .Performance of the developed network was tested with the help of (1) drawing a scatter diagram of estimated versus target values (2) evaluating correlation coefficient, r given bywhere Xn is network output, Xt is target output, n is number of test patterns.(4) Determining coef

55、ficient of efficiency given by:3. DataThe observations reported by Bayram and Larsen 5 on the pile scouring are used for training as well as testing. This is because they are relatively exhaustive and pertains to direct prototype observations while most of the reported studies are based on laborator

56、y experiments. The data were gathered at a 200 m long research pier constructed at Ajigaura Beach on the Pacific coast of Japan. The pier projected out from the shore into the sea and was supported by nine groups of vertical piles, each having four piles of 0.6m diameter placed at 2.67m c/c as shown

57、 in Fig. 4(a). Each pile group was 30m apart from each other. Dimensions of the scour hole around a pile group were available along with corresponding environmental parameters through 56 weekly surveys made during the period of August 1984 to September 1989. This pile group was located at the seawar

58、d end of the pier, away from the action of breaking waves and where oscillatory waves prevailed. The scour dimensions corresponded to the equilibrium state and wave height and period values belonged to preceding weekly averages. The average grain size of the sediment was 0.2 mm. Even though the cent

59、re to centre distance between the piles in the group was larger (around 4.5 times the individual pile diameter) the group had a common scour hole of dimension s (depth) and L w (width) as shown in Fig. 4(b).Past studies on scour 3 involved evaluation of the scour dimensions using certain dimensionle

60、ss groups of individual parameters. They are Reynolds number, Re; Keulegan and Carpenter number, Kc, Shields parameter, u; and, Sediment number, Ns. The expressions for these numbers are given below:where U is maximum water particle velocity induced by the wave motion (determined by the linear wave

61、theory herein), D is pile diameter, V is kinematics viscosity of the sea water, T is wave period, V is undisturbed bed shear velocity given by:is wave friction factor, S is specific gravity of the sediment; g is acceleration due to gravity, is mean diameter of the sediment. The ranges of different p

62、arameters involved in this study are given in Table 1.Although previous studies incorporated these dimensionless group of parameters to evaluate scour hole dimensions, some investigators working on neural networks 8 have earlier reported that use of constitutive raw parameters in place of their grou

63、ps yields better results because of the increased flexibility in fitting achieved in that way. It was therefore decided to develop a network with input of causative raw variables, namely, H; T; and, water depth, d; which vary from time to time at a given location and also that of D; S; f; and, f; wh

64、ich remain fixed. It was however subsequently noticed that the latter constants do not contribute much to the outcome and instead of that it is sufficient to account for them by inputting only a single parameter D: The effect of all four constants might have been indirectly present in the weights an

65、d bias of the network. Also this might have resulted in avoiding overfitting errors during network training. Further, the scour dimensions are generally found to be independent of the sediment properties 5. A network consisting of four inputs, namely, H; T; d; and, D was therefore first considered to predict the dimensionless scour depth, S/D:As per