经过一个多月的努力，终于完成了BP网络，参考的资料为：

1、Training feed-forward networks with the Marquardt algorithm

2、The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems

3、Neural Network Design

4、http://deeplearning.stanford.edu/wiki/index.php/UFLDL%E6%95%99%E7%A8%8B 中介绍的神经网络部分

以下给出Python脚本：

 

import numpy as npfrom math import exp, powfrom mpl_toolkits.mplot3d import Axes3Dimport matplotlib.pyplot as pltimport sysimport copyfrom scipy.linalg import norm, pinvclass Layer:    def __init__(self,w, b, neure_number, transfer_function, layer_index):        self.transfer_function = transfer_function        self.neure_number = neure_number        self.layer_index = layer_index        self.w = w        self.b = b        class NetStruct:    def __init__(self, x, y, hidden_layers, activ_fun_list, performance_function = 'mse'):        if len(hidden_layers) == len(activ_fun_list):            activ_fun_list.append('line')        self.active_fun_list = activ_fun_list        self.performance_function = performance_function        x = np.array(x)        y = np.array(y)        if(x.shape[1] != y.shape[1]):            print 'The dimension of x and y are not same.'            sys.exit()        self.x = x        self.y = y        input_eles = self.x.shape[0]        output_eles = self.y.shape[0]        tmp = []        tmp.append(input_eles)        tmp.extend(hidden_layers)        tmp.append(output_eles)        self.hidden_layers = np.array(tmp)        self.layer_num = len(self.hidden_layers)        self.layers = []        for i in range(0, len(self.hidden_layers)):                        if i == 0:                self.layers.append(Layer([],[],\                                        self.hidden_layers[i], 'none', i))                 continue            f = self.hidden_layers[i - 1]            s = self.hidden_layers[i]             self.layers.append(Layer(np.random.randn(s, f),np.random.randn(s, 1),\                                        self.hidden_layers[i], activ_fun_list[i-1], i))     class Train:    def __init__(self, net_struct, mu = 1e-3, beta = 10, iteration = 100, tol = 0.1):        self.net_struct = net_struct        self.mu = mu        self.beta = beta        self.iteration = iteration        self.tol = tol    def train(self, method = 'lm'):        if(method == 'lm'):            self.lm()    def sim(self, x):        self.net_struct.x = x        self.forward()        layer_num = len(self.net_struct.layers)        predict = self.net_struct.layers[layer_num - 1].output_val        return predict    def actFun(self, z, activ_type = 'sigm'):        if activ_type == 'sigm':                        f = 1.0 / (1.0 + np.exp(-z))        elif activ_type == 'tanh':            f = (np.exp(z) + np.exp(-z)) / (np.exp(z) + np.exp(-z))        elif activ_type == 'radb':            f = np.exp(-z * z)        elif activ_type == 'line':            f = z        return f    def actFunGrad(self, z, activ_type = 'sigm'):        if activ_type == 'sigm':            grad = self.actFun(z, activ_type) * (1.0 - self.actFun(z, activ_type))        elif activ_type == 'tanh':            grad = 1.0 - self.actFun(z, activ_type) * self.actFun(z, activ_type)        elif activ_type == 'radb':            grad = -2.0 * z * self.actFun(z, activ_type)        elif activ_type == 'line':            m = z.shape[0]            n = z.shape[1]            grad = np.ones((m, n))        return grad    def forward(self):        layer_num = len(self.net_struct.layers)        for i in range(0, layer_num):            if i == 0:                curr_layer = self.net_struct.layers[i]                curr_layer.input_val = self.net_struct.x                curr_layer.output_val = self.net_struct.x                continue            before_layer = self.net_struct.layers[i - 1]            curr_layer = self.net_struct.layers[i]            curr_layer.input_val = curr_layer.w.dot(before_layer.output_val) + curr_layer.b            curr_layer.output_val = self.actFun(curr_layer.input_val,                                                 self.net_struct.active_fun_list[i - 1])    def backward(self):        layer_num = len(self.net_struct.layers)        last_layer = self.net_struct.layers[layer_num - 1]        last_layer.error = -self.actFunGrad(last_layer.input_val,                                            self.net_struct.active_fun_list[layer_num - 2])        layer_index = range(1, layer_num - 1)        layer_index.reverse()        for i in layer_index:            curr_layer = self.net_struct.layers[i]            curr_layer.error = (last_layer.w.transpose().dot(last_layer.error)) \                      * self.actFunGrad(curr_layer.input_val,self.net_struct.active_fun_list[i - 1])            last_layer = curr_layer    def parDeriv(self):        layer_num = len(self.net_struct.layers)        for i in range(1, layer_num):            befor_layer = self.net_struct.layers[i - 1]            befor_input_val = befor_layer.output_val.transpose()            curr_layer = self.net_struct.layers[i]            curr_error = curr_layer.error            curr_error = curr_error.reshape(curr_error.shape[0]*curr_error.shape[1], 1, order='F')            row =  curr_error.shape[0]            col = befor_input_val.shape[1]            a = np.zeros((row, col))            num = befor_input_val.shape[0]            neure_number = curr_layer.neure_number            for i in range(0, num):                a[neure_number*i:neure_number*i + neure_number,:] = \                 np.repeat([befor_input_val[i,:]],neure_number,axis = 0)            tmp_w_par_deriv = curr_error * a            curr_layer.w_par_deriv = np.zeros((num, befor_layer.neure_number * curr_layer.neure_number))            for i in range(0, num):                tmp = tmp_w_par_deriv[neure_number*i:neure_number*i + neure_number,:]                tmp = tmp.reshape(tmp.shape[0] * tmp.shape[1], order='C')                curr_layer.w_par_deriv[i, :] = tmp            curr_layer.b_par_deriv = curr_layer.error.transpose()    def jacobian(self):        layers = self.net_struct.hidden_layers        row = self.net_struct.x.shape[1]        col = 0        for i in range(0, len(layers) - 1):            col = col + layers[i] * layers[i + 1] + layers[i + 1]        j = np.zeros((row, col))        layer_num = len(self.net_struct.layers)        index = 0        for i in range(1, layer_num):            curr_layer = self.net_struct.layers[i]            w_col = curr_layer.w_par_deriv.shape[1]            b_col = curr_layer.b_par_deriv.shape[1]            j[:, index : index + w_col] = curr_layer.w_par_deriv            index = index + w_col            j[:, index : index + b_col] = curr_layer.b_par_deriv            index = index + b_col        return j    def gradCheck(self):        W1 = self.net_struct.layers[1].w        b1 = self.net_struct.layers[1].b        n = self.net_struct.layers[1].neure_number        W2 = self.net_struct.layers[2].w        b2 = self.net_struct.layers[2].b        x = self.net_struct.x        p = []        p.extend(W1.reshape(1,W1.shape[0]*W1.shape[1],order = 'C')[0])        p.extend(b1.reshape(1,b1.shape[0]*b1.shape[1],order = 'C')[0])        p.extend(W2.reshape(1,W2.shape[0]*W2.shape[1],order = 'C')[0])        p.extend(b2.reshape(1,b2.shape[0]*b2.shape[1],order = 'C')[0])        old_p = p        jac = []        for i in range(0, x.shape[1]):            xi = np.array([x[:,i]])            xi = xi.transpose()            ji = []            for j in range(0, len(p)):                W1 = np.array(p[0:2*n]).reshape(n,2,order='C')                b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')                W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')                b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')                                z2 = W1.dot(xi) + b1                a2 = self.actFun(z2)                z3 = W2.dot(a2) + b2                h1 = self.actFun(z3)                p[j] = p[j] + 0.00001                W1 = np.array(p[0:2*n]).reshape(n,2,order='C')                b1 = np.array(p[2*n:2*n+n]).reshape(n,1,order='C')                W2 = np.array(p[3*n:4*n]).reshape(1,n,order='C')                b2 = np.array(p[4*n:4*n+1]).reshape(1,1,order='C')                                z2 = W1.dot(xi) + b1                a2 = self.actFun(z2)                z3 = W2.dot(a2) + b2                h = self.actFun(z3)                g = (h[0][0]-h1[0][0])/0.00001                ji.append(g)            jac.append(ji)            p = old_p        return jac    def jjje(self):        layer_number = self.net_struct.layer_num        e = self.net_struct.y - \          self.net_struct.layers[layer_number - 1].output_val        e = e.transpose()        j = self.jacobian()        #check gradient        #j1 = -np.array(self.gradCheck())        #jk = j.reshape(1,j.shape[0]*j.shape[1])        #jk1 = j1.reshape(1,j1.shape[0]*j1.shape[1])        #plt.plot(jk[0])        #plt.plot(jk1[0],'.')        #plt.show()        jj = j.transpose().dot(j)        je = -j.transpose().dot(e)        return[jj, je]    def lm(self):        mu = self.mu        beta = self.beta        iteration = self.iteration        tol = self.tol        y = self.net_struct.y        self.forward()        pred =  self.net_struct.layers[self.net_struct.layer_num - 1].output_val        pref = self.perfermance(y, pred)        for i in range(0, iteration):            print 'iter:',i, 'error:', pref            #1) step 1:             if(pref < tol):                break            #2) step 2:            self.backward()            self.parDeriv()            [jj, je] = self.jjje()            while(1):                #3) step 3:                 A = jj + mu * np.diag(np.ones(jj.shape[0]))                delta_w_b = pinv(A).dot(je)                #4) step 4:                old_net_struct = copy.deepcopy(self.net_struct)                self.updataNetStruct(delta_w_b)                self.forward()                pred1 =  self.net_struct.layers[self.net_struct.layer_num - 1].output_val                pref1 = self.perfermance(y, pred1)                if (pref1 < pref):                    mu = mu / beta                    pref = pref1                    break                mu = mu * beta                self.net_struct = copy.deepcopy(old_net_struct)    def updataNetStruct(self, delta_w_b):        layer_number = self.net_struct.layer_num        index = 0        for i in range(1, layer_number):            before_layer = self.net_struct.layers[i - 1]            curr_layer = self.net_struct.layers[i]            w_num = before_layer.neure_number * curr_layer.neure_number            b_num = curr_layer.neure_number            w = delta_w_b[index : index + w_num]            w = w.reshape(curr_layer.neure_number, before_layer.neure_number, order='C')            index = index + w_num            b = delta_w_b[index : index + b_num]            index = index + b_num            curr_layer.w += w            curr_layer.b += b    def perfermance(self, y, pred):        error = y - pred        return norm(error) / len(y)      def plotSamples(self, n = 40):        x = np.array([np.linspace(0, 3, n)])        x = x.repeat(n, axis = 0)        y = x.transpose()        z = np.zeros((n, n))        for i in range(0, x.shape[0]):            for j in range(0, x.shape[1]):                z[i][j] = self.sampleFun(x[i][j], y[i][j])        fig = plt.figure()        ax = fig.gca(projection='3d')        surf = ax.plot_surface(x, y, z, cmap='autumn', cstride=2, rstride=2)        ax.set_xlabel("X-Label")        ax.set_ylabel("Y-Label")        ax.set_zlabel("Z-Label")        plt.show()def sinSamples(n):        x = np.array([np.linspace(-0.5, 0.5, n)])        #x = x.repeat(n, axis = 0)        y = x + 0.2        z = np.zeros((n, 1))        for i in range(0, x.shape[1]):            z[i] = np.sin(x[0][i] * y[0][i])        X = np.zeros((n, 2))        n = 0        for xi, yi in zip(x.transpose(), y.transpose()):            X[n][0] = xi            X[n][1] = yi            n = n + 1        return X,zdef peaksSamples(n):    x = np.array([np.linspace(-3, 3, n)])    x = x.repeat(n, axis = 0)    y = x.transpose()    z = np.zeros((n, n))    for i in range(0, x.shape[0]):        for j in range(0, x.shape[1]):            z[i][j] = sampleFun(x[i][j], y[i][j])    X = np.zeros((n*n, 2))    x_list = x.reshape(n*n,1 )    y_list = y.reshape(n*n,1)    z_list = z.reshape(n*n,1)    n = 0    for xi, yi in zip(x_list, y_list):        X[n][0] = xi        X[n][1] = yi        n = n + 1        return X,z_list.transpose()def sampleFun(x, y):    z =  3*pow((1-x),2) * exp(-(pow(x,2)) - pow((y+1),2)) \   - 10*(x/5 - pow(x, 3) - pow(y, 5)) * exp(-pow(x, 2) - pow(y, 2)) \   - 1/3*exp(-pow((x+1), 2) - pow(y, 2))     return zif __name__ == '__main__':        hidden_layers = [10,10] #设置网络层数，共两层，每层10个神经元    activ_fun_list = ['sigm','sigm']#设置隐层的激活函数类型，可以设置为tanh,radb，tanh,line类型，如果不显式的设置最后一层为line     [X, z] = peaksSamples(20) #产生训练数据点    X = X.transpose()    bp = NetStruct(X, z, hidden_layers, activ_fun_list) #初始化网络信息    tr = Train(bp) #初始化训练网络的类    tr.train() #训练    [XX, z0] = peaksSamples(40) #产生测试数据    XX = XX.transpose()    z1 = tr.sim(XX) #用训练好的神经网络预测数据，z1为预测结果        fig  = plt.figure()    ax = fig.add_subplot(111)    ax.plot(z0[0]) #真值    ax.plot(z1[0],'r.') #预测值    plt.legend((r'real data', r'predict data'))    plt.show()

以上代码计算的结果如下图，由于初始值等原因的影响偶尔收敛效果会变差，不过大多数时候都可以收敛到下图的结果，以后再改进，欢迎指正。