\n| \n E<\/p>\n<\/td>\n | \n 0.8235<\/p>\n<\/td>\n | \n 0.6471<\/p>\n<\/td>\n | \n 0.8235<\/p>\n<\/td>\n | \n 0.7647<\/p>\n<\/td>\n | \n 0.6652<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u6240\u6709\u8f93\u5165\u5df2\u6807\u51c6\u5316\u5230[0,1]，\u8d8a\u5927\u8868\u793a\u6027\u80fd\u8d8a\u597d（\u6210\u672c\u53cd\u5411\u5904\u7406）\u3002<\/p>\n \u56db\u3001C\u8bed\u8a00\u5b9e\u73b0<\/h2>\n4.1 \u6838\u5fc3\u8bbe\u8ba1\u601d\u8def<\/h3>\nC\u7248\u672c\u9762\u5411\u5de5\u7a0b\u5d4c\u5165，\u8bbe\u8ba1\u8981\u70b9：<\/p>\n \n- \u6240\u6709\u7ef4\u5ea6\u7528\u5b8f\u5b9a\u4e49，\u7f16\u8bd1\u671f\u786e\u5b9a<\/li>\n
- \u6279\u91cf\u8bad\u7ec3，\u77e9\u9635\u8fd0\u7b97\u5c55\u5f00\u4e3a\u591a\u5c42\u5faa\u73af<\/li>\n
- \u540c\u65f6\u652f\u6301\u63a7\u5236\u53f0\u548c\u6587\u4ef6\u8f93\u51fa<\/li>\n
- \u56fa\u5b9a\u968f\u673a\u79cd\u5b50\u4fdd\u8bc1\u53ef\u590d\u73b0<\/li>\n<\/ul>\n
4.2 \u5b8c\u6574\u4ee3\u7801<\/h3>\n\/**
\n* @file : Alg_11_BPNN.h
\n* @brief : BP\u795e\u7ecf\u7f51\u7edc\u6a21\u578b
\n* @details : \u57fa\u4e8e\u8bef\u5dee\u53cd\u5411\u4f20\u64ad\u7684\u591a\u5c42\u524d\u9988\u76d1\u7763\u5b66\u4e60\u7f51\u7edc，\u7528\u4e8e\u975e\u7ebf\u6027\u62df\u5408\u4e0e\u6548\u80fd\u9884\u6d4b
\n* @author : \u4e09\u73af\u4e0a\u7684\u9a91\u58eb
\n* @date : 2026-05
\n* @version : v1.0
\n* @note : \u9700\u76d1\u7763\u4fe1\u53f7（\u5982\u4e13\u5bb6\u7efc\u5408\u6548\u80fd\u503c）\u8fdb\u884c\u8bad\u7ec3
\n* @warning : \u968f\u673a\u521d\u59cb\u5316\u6613\u9677\u5c40\u90e8\u6781\u5c0f，\u7ed3\u679c\u5b58\u5728\u4e00\u5b9a\u6ce2\u52a8\u6027
\n*\/<\/p>\n #include <stdio.h>
\n#include <stdlib.h>
\n#include <math.h>
\n#include <time.h><\/p>\n #define INNUM 4 \/\/ \u8f93\u5165\u7ef4\u6570
\n#define HIDNUM 6 \/\/ \u9690\u5c42\u8282\u70b9\u6570
\n#define OUTNUM 1 \/\/ \u8f93\u51fa\u7ef4\u6570
\n#define SAMPLE 5 \/\/ \u6837\u672c\u6570
\n#define EPOCHS 6000
\n#define LR 0.05
\n#define EPS 1e-8
\n#define RESULTPATH "..\/..\/..\/006_\u6548\u80fd\u8bc4\u4f30\u7ed3\u679c\u6570\u636e\u5305\/11_bpnn_result.txt"<\/p>\n \/* \u6fc0\u6d3b\u51fd\u6570 *\/
\nstatic inline double sigmoid2(double x)
\n{
\nreturn 1.0 \/ (1.0 + exp(-x));
\n}<\/p>\n \/* \u968f\u673a\u6570\u521d\u59cb\u5316（Xavier） *\/
\nvoid init_weights(double *w, int fan_in, int fan_out)
\n{
\nsrand(2026);
\ndouble limit = sqrt(6.0 \/ (fan_in + fan_out));
\nfor (int i = 0; i < fan_in * fan_out; i++)
\n{
\nw[i] = ((double)rand() \/ RAND_MAX * 2.0 – 1.0) * limit;
\n}
\n}<\/p>\n \/* BP\u8bad\u7ec3\u4e0e\u9884\u6d4b *\/
\nvoid bp_train_predict(
\ndouble P[INNUM][SAMPLE],
\ndouble T[SAMPLE],
\ndouble W1[HIDNUM][INNUM],
\ndouble b1[HIDNUM],
\ndouble W2[OUTNUM][HIDNUM],
\ndouble b2[OUTNUM],
\ndouble Y_pred[SAMPLE],
\ndouble loss_rec[EPOCHS]
\n)
\n{
\nint epoch, s, i, j;
\ndouble Z1[HIDNUM][SAMPLE];
\ndouble A1[HIDNUM][SAMPLE];
\ndouble Z2[OUTNUM][SAMPLE];
\ndouble A2[OUTNUM][SAMPLE];
\ndouble E[OUTNUM][SAMPLE];<\/p>\n double dZ2[OUTNUM][SAMPLE];
\ndouble dW2[OUTNUM][HIDNUM];
\ndouble db2[OUTNUM];<\/p>\n double dA1[HIDNUM][SAMPLE];
\ndouble dZ1[HIDNUM][SAMPLE];
\ndouble dW1[HIDNUM][INNUM];
\ndouble db1[HIDNUM];<\/p>\n for (epoch = 0; epoch < EPOCHS; epoch++)
\n{<\/p>\n \/* ———- \u524d\u5411\u4f20\u64ad ———- *\/
\nfor (s = 0; s < SAMPLE; s++)
\n{
\nfor (i = 0; i < HIDNUM; i++)
\n{
\nZ1[i][s] = b1[i];
\nfor (j = 0; j < INNUM; j++)
\nZ1[i][s] += W1[i][j] * P[j][s];
\nA1[i][s] = sigmoid2(Z1[i][s]);
\n}
\n}<\/p>\n for (s = 0; s < SAMPLE; s++)
\n{
\nfor (i = 0; i < OUTNUM; i++)
\n{
\nZ2[i][s] = b2[i];
\nfor (j = 0; j < HIDNUM; j++)
\nZ2[i][s] += W2[i][j] * A1[j][s];
\nA2[i][s] = Z2[i][s]; \/\/ purelin
\n}
\n}<\/p>\n \/* ———- \u8bef\u5dee ———- *\/
\ndouble loss = 0.0;
\nfor (s = 0; s < SAMPLE; s++)
\n{
\nfor (i = 0; i < OUTNUM; i++)
\n{
\nE[i][s] = A2[i][s] – T[s];
\nloss += E[i][s] * E[i][s];
\n}
\n}
\nloss_rec[epoch] = loss \/ (2.0 * SAMPLE);<\/p>\n \/* ———- \u53cd\u5411\u4f20\u64ad ———- *\/
\n\/* \u8f93\u51fa\u5c42 *\/
\nfor (s = 0; s < SAMPLE; s++)
\nfor (i = 0; i < OUTNUM; i++)
\ndZ2[i][s] = E[i][s]; \/\/ purelin\u5bfc\u6570=1<\/p>\n for (i = 0; i < OUTNUM; i++)
\nfor (j = 0; j < HIDNUM; j++)
\n{
\ndW2[i][j] = 0.0;
\nfor (s = 0; s < SAMPLE; s++)
\ndW2[i][j] += dZ2[i][s] * A1[j][s];
\ndW2[i][j] \/= SAMPLE;
\n}<\/p>\n for (i = 0; i < OUTNUM; i++)
\n{
\ndb2[i] = 0.0;
\nfor (s = 0; s < SAMPLE; s++)
\ndb2[i] += dZ2[i][s];
\ndb2[i] \/= SAMPLE;
\n}<\/p>\n \/* \u9690\u5c42 *\/
\nfor (s = 0; s < SAMPLE; s++)
\nfor (i = 0; i < HIDNUM; i++)
\n{
\ndA1[i][s] = 0.0;
\nfor (j = 0; j < OUTNUM; j++)
\ndA1[i][s] += W2[j][i] * dZ2[j][s];
\ndZ1[i][s] = dA1[i][s] * A1[i][s] * (1.0 – A1[i][s]);
\n}<\/p>\n for (i = 0; i < HIDNUM; i++)
\nfor (j = 0; j < INNUM; j++)
\n{
\ndW1[i][j] = 0.0;
\nfor (s = 0; s < SAMPLE; s++)
\ndW1[i][j] += dZ1[i][s] * P[j][s];
\ndW1[i][j] \/= SAMPLE;
\n}<\/p>\n for (i = 0; i < HIDNUM; i++)
\n{
\ndb1[i] = 0.0;
\nfor (s = 0; s < SAMPLE; s++)
\ndb1[i] += dZ1[i][s];
\ndb1[i] \/= SAMPLE;
\n}<\/p>\n \/* ———- \u53c2\u6570\u66f4\u65b0 ———- *\/
\nfor (i = 0; i < OUTNUM; i++)
\n{
\nfor (j = 0; j < HIDNUM; j++)
\nW2[i][j] -= LR * dW2[i][j];
\nb2[i] -= LR * db2[i];
\n}<\/p>\n for (i = 0; i < HIDNUM; i++)
\n{
\nfor (j = 0; j < INNUM; j++)
\nW1[i][j] -= LR * dW1[i][j];
\nb1[i] -= LR * db1[i];
\n}<\/p>\n if (epoch % 500 == 0)
\nprintf("Epoch %4d | MSE = %.6f\\\\n", epoch, loss_rec[epoch]);
\n}<\/p>\n \/* ———- \u9884\u6d4b\u8f93\u51fa ———- *\/
\nfor (s = 0; s < SAMPLE; s++)
\n{
\ndouble sum = b2[0];
\nfor (j = 0; j < HIDNUM; j++)
\n{
\ndouble a1 = sigmoid2(W1[j][0] * P[0][s] +
\nW1[j][1] * P[1][s] +
\nW1[j][2] * P[2][s] +
\nW1[j][3] * P[3][s] + b1[j]);
\nsum += W2[0][j] * a1;
\n}
\nY_pred[s] = sum;
\n}
\n}
\n\/* ================= BPNN \u7ed3\u679c\u8f93\u51fa（\u63a7\u5236\u53f0） ================= *\/
\nvoid print_bpnn_results(
\ndouble T[SAMPLE],
\ndouble Y_pred[SAMPLE],
\ndouble loss_rec[EPOCHS]
\n)
\n{
\nint i;
\ndouble mse = 0.0, mae = 0.0, rmse = 0.0;
\ndouble mean = 0.0, var = 0.0, r2 = 0.0;<\/p>\n \/* ———- 1. \u8bad\u7ec3\u8bef\u5dee\u7edf\u8ba1 ———- *\/
\nfor (i = 0; i < SAMPLE; i++)
\nmean += T[i];
\nmean \/= SAMPLE;<\/p>\n for (i = 0; i < SAMPLE; i++)
\n{
\ndouble err = Y_pred[i] – T[i];
\nmse += err * err;
\nmae += fabs(err);
\nvar += (T[i] – mean) * (T[i] – mean);
\n}
\nmse \/= SAMPLE;
\nmae \/= SAMPLE;
\nrmse = sqrt(mse);
\nr2 = (var < EPS) ? 0.0 : (1.0 – mse * SAMPLE \/ var);<\/p>\n printf("\\\\n========== BP\u795e\u7ecf\u7f51\u7edc\u6548\u80fd\u8bc4\u4f30\u7ed3\u679c ==========\\\\n");
\nprintf("\u7f51\u7edc\u7ed3\u6784: %d-%d-%d (\u8f93\u5165-\u9690\u5c42-\u8f93\u51fa)\\\\n", INNUM, HIDNUM, OUTNUM);
\nprintf("\u8bad\u7ec3\u8f6e\u6570: %d, \u5b66\u4e60\u7387: %.4f\\\\n\\\\n", EPOCHS, LR);<\/p>\n \/* ———- 2. \u6837\u672c\u9884\u6d4b\u660e\u7ec6 ———- *\/
\nprintf("———- 1. \u6837\u672c\u9884\u6d4b\u660e\u7ec6 ———-\\\\n");
\nprintf("%-8s %10s %10s %12s %12s\\\\n",
\n"\u65b9\u6848", "\u771f\u5b9e\u503c", "\u9884\u6d4b\u503c", "\u7edd\u5bf9\u8bef\u5dee", "\u76f8\u5bf9\u8bef\u5dee");
\nconst char *labels[SAMPLE] = { "A", "B", "C", "D", "E" };
\nfor (i = 0; i < SAMPLE; i++)
\n{
\ndouble abs_err = fabs(Y_pred[i] – T[i]);
\ndouble rel_err = (T[i] != 0.0) ? abs_err \/ T[i] : 0.0;
\nprintf("%-8s %10.4f %10.4f %12.4f %11.2f%%\\\\n",
\nlabels[i], T[i], Y_pred[i], abs_err, rel_err * 100.0);
\n}
\nprintf("\\\\n");<\/p>\n \/* ———- 3. \u6a21\u578b\u6027\u80fd\u6307\u6807 ———- *\/
\nprintf("———- 2. \u6a21\u578b\u6027\u80fd\u6307\u6807 ———-\\\\n");
\nprintf("MSE = %.6f\\\\n", mse);
\nprintf("RMSE = %.6f\\\\n", rmse);
\nprintf("MAE = %.6f\\\\n", mae);
\nprintf("R\u00b2 = %.6f\\\\n", r2);
\nprintf("\\\\n");<\/p>\n \/* ———- 4. \u8bad\u7ec3\u6536\u655b\u60c5\u51b5 ———- *\/
\nprintf("———- 3. \u8bad\u7ec3\u6536\u655b\u60c5\u51b5 ———-\\\\n");
\nprintf("\u521d\u59cb\u635f\u5931: %.6f\\\\n", loss_rec[0]);
\nprintf("\u6700\u7ec8\u635f\u5931: %.6f\\\\n", loss_rec[EPOCHS – 1]);
\nprintf("\u635f\u5931\u4e0b\u964d: %.6f\\\\n", loss_rec[0] – loss_rec[EPOCHS – 1]);<\/p>\n \/* ———- 5. \u6a21\u578b\u8bc4\u4ef7\u7ed3\u8bba ———- *\/
\nprintf("\\\\n———- 4. \u6a21\u578b\u8bc4\u4ef7\u7ed3\u8bba ———-\\\\n");
\nif (r2 > 0.95)
\nprintf("\u2705 \u6a21\u578b\u62df\u5408\u4f18\u5ea6\u4f18\u79c0 (R\u00b2 > 0.95)\\\\n");
\nelse if (r2 > 0.85)
\nprintf("\u2705 \u6a21\u578b\u62df\u5408\u4f18\u5ea6\u826f\u597d (R\u00b2 > 0.85)\\\\n");
\nelse if (r2 > 0.70)
\nprintf("\u26a0\ufe0f \u6a21\u578b\u62df\u5408\u4e00\u822c，\u5efa\u8bae\u589e\u52a0\u6837\u672c\u6216\u8c03\u6574\u7ed3\u6784\\\\n");
\nelse
\nprintf("\u274c \u6a21\u578b\u62df\u5408\u8f83\u5dee，\u9700\u91cd\u65b0\u8bbe\u8ba1\u7f51\u7edc\u6216\u68c0\u67e5\u6570\u636e\\\\n");<\/p>\n if (rmse < 0.05)
\nprintf("\u2705 \u9884\u6d4b\u7cbe\u5ea6\u6ee1\u8db3\u5de5\u7a0b\u9700\u6c42 (RMSE < 0.05)\\\\n");
\nelse
\nprintf("\u26a0\ufe0f \u9884\u6d4b\u8bef\u5dee\u504f\u5927，\u5efa\u8bae\u5ef6\u957f\u8bad\u7ec3\u6216\u8c03\u6574\u5b66\u4e60\u7387\\\\n");<\/p>\n printf("==========================================\\\\n");
\n}<\/p>\n \/* ================= BPNN \u7ed3\u679c\u8f93\u51fa（\u6587\u4ef6\u7248） ================= *\/
\n\/* ================= BP \u795e\u7ecf\u7f51\u7edc\u7ed3\u679c\u8f93\u51fa（\u6587\u4ef6\u7248） ================= *\/
\nvoid print_bpnn_results_to_file(
\ndouble T[SAMPLE],
\ndouble Y_pred[SAMPLE],
\ndouble loss_rec[EPOCHS],
\nFILE *fp
\n)
\n{
\nint i;
\ndouble mse = 0.0, mae = 0.0, rmse = 0.0;
\ndouble mean = 0.0, var = 0.0, r2 = 0.0;<\/p>\n \/* ———- 1. \u7edf\u8ba1\u6307\u6807\u8ba1\u7b97 ———- *\/
\nfor (i = 0; i < SAMPLE; i++) mean += T[i];
\nmean \/= SAMPLE;<\/p>\n for (i = 0; i < SAMPLE; i++)
\n{
\ndouble err = Y_pred[i] – T[i];
\nmse += err * err;
\nmae += fabs(err);
\nvar += (T[i] – mean) * (T[i] – mean);
\n}
\nmse \/= SAMPLE;
\nmae \/= SAMPLE;
\nrmse = sqrt(mse);
\nr2 = (var < EPS) ? 0.0 : (1.0 – mse * SAMPLE \/ var);<\/p>\n \/* ———- 2. \u6587\u4ef6\u5934\u4fe1\u606f ———- *\/
\nfprintf(fp, "\\\\n========== BP \u795e\u7ecf\u7f51\u7edc\u6548\u80fd\u8bc4\u4f30\u7ed3\u679c ==========\\\\n");
\nfprintf(fp, "\u7f51\u7edc\u7ed3\u6784: %d-%d-%d\\\\n", INNUM, HIDNUM, OUTNUM);
\nfprintf(fp, "\u8bad\u7ec3\u8f6e\u6570: %d, \u5b66\u4e60\u7387: %.4f\\\\n\\\\n", EPOCHS, LR);<\/p>\n \/* ———- 3. \u6837\u672c\u9884\u6d4b\u660e\u7ec6 ———- *\/
\nfprintf(fp, "———- 1. \u6837\u672c\u9884\u6d4b\u660e\u7ec6 ———-\\\\n");
\nfprintf(fp, "%-8s %10s %10s %12s\\\\n",
\n"\u65b9\u6848", "\u771f\u5b9e\u503c", "\u9884\u6d4b\u503c", "\u7edd\u5bf9\u8bef\u5dee");
\nconst char *labels[SAMPLE] = { "A", "B", "C", "D", "E" };
\nfor (i = 0; i < SAMPLE; i++)
\n{
\nfprintf(fp, "%-8s %10.4f %10.4f %12.4f\\\\n",
\nlabels[i], T[i], Y_pred[i], fabs(Y_pred[i] – T[i]));
\n}
\nfprintf(fp, "\\\\n");<\/p>\n \/* ———- 4. \u6a21\u578b\u6027\u80fd\u6307\u6807 ———- *\/
\nfprintf(fp, "———- 2. \u6a21\u578b\u6027\u80fd\u6307\u6807 ———-\\\\n");
\nfprintf(fp, "MSE = %.6f\\\\n", mse);
\nfprintf(fp, "RMSE = %.6f\\\\n", rmse);
\nfprintf(fp, "MAE = %.6f\\\\n", mae);
\nfprintf(fp, "R\u00b2 = %.6f\\\\n", r2);
\nfprintf(fp, "\\\\n");<\/p>\n \/* ———- 5. \u8bad\u7ec3\u6536\u655b\u60c5\u51b5 ———- *\/
\nfprintf(fp, "———- 3. \u8bad\u7ec3\u6536\u655b\u60c5\u51b5 ———-\\\\n");
\nfprintf(fp, "\u521d\u59cb\u635f\u5931: %.6f\\\\n", loss_rec[0]);
\nfprintf(fp, "\u6700\u7ec8\u635f\u5931: %.6f\\\\n", loss_rec[EPOCHS – 1]);
\nfprintf(fp, "\u635f\u5931\u4e0b\u964d\u91cf: %.6f\\\\n", loss_rec[0] – loss_rec[EPOCHS – 1]);<\/p>\n fprintf(fp, "==============================================\\\\n");
\n}<\/p>\n \/*************************\u8c03\u7528\u4f8b\u5b50**************************\/
\nvoid testBPNN()
\n{
\n\/* \u6570\u636e（\u4e0eMATLAB\u5b8c\u5168\u4e00\u81f4） *\/
\ndouble P[INNUM][SAMPLE] =
\n{
\n{ 0.5294, 0.1765, 0.4118, 1.0000 },
\n{ 0.2353, 0.3529, 0.1765, 0.4118 },
\n{ 1.0000, 0.0000, 1.0000, 0.0000 },
\n{ 0.0000, 0.4706, 0.0000, 0.4118 }
\n};<\/p>\n double T[SAMPLE] = { 0.6218, 0.4873, 0.7125, 0.8031, 0.6652 };<\/p>\n double W1[HIDNUM][INNUM];
\ndouble b1[HIDNUM] = { 0 };
\ndouble W2[OUTNUM][HIDNUM];
\ndouble b2[OUTNUM] = { 0 };<\/p>\n double Y_pred[SAMPLE];
\ndouble loss_rec[EPOCHS];<\/p>\n \/* \u521d\u59cb\u5316 *\/
\ninit_weights(&W1[0][0], INNUM, HIDNUM);
\ninit_weights(&W2[0][0], HIDNUM, OUTNUM);<\/p>\n \/* \u8bad\u7ec3 *\/
\nbp_train_predict(P, T, W1, b1, W2, b2, Y_pred, loss_rec);<\/p>\n \/* \u63a7\u5236\u53f0\u8f93\u51fa *\/
\nprint_bpnn_results(T, Y_pred, loss_rec);<\/p>\n \/* \u6587\u4ef6\u8f93\u51fa *\/
\nFILE *fp = fopen(RESULTPATH, "w");
\nif (fp != NULL)
\n{
\nprint_bpnn_results_to_file(T, Y_pred, loss_rec, fp);
\nfclose(fp);
\nprintf("[INFO] BPNN\u8bc4\u4f30\u7ed3\u679c\u5df2\u5199\u5165: %s\\\\n", RESULTPATH);
\n}
\nelse
\n{
\nprintf("[ERROR] \u65e0\u6cd5\u6253\u5f00\u6587\u4ef6: %s\\\\n", RESULTPATH);
\n}
\n} <\/p>\n 4.3 \u5173\u952e\u5b9e\u73b0\u7ec6\u8282<\/h3>\n\n\n\u8981\u70b9<\/p>\n \u8bf4\u660e<\/p>\n<\/tr>\n \n\n| \n \u6570\u636e\u5b58\u50a8<\/p>\n<\/td>\n | \n P[INNUM][SAMPLE] \u884c=\u7279\u5f81\u3001\u5217=\u6837\u672c，\u4e0eMATLAB\u8f6c\u7f6e\u540e\u5bf9\u5e94<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u77e9\u9635\u8fd0\u7b97<\/p>\n<\/td>\n | \n \u5c55\u5f00\u4e3afor\u5faa\u73af，\u65e0\u5916\u90e8\u4f9d\u8d56，\u9002\u5408\u5d4c\u5165\u5f0f\u73af\u5883<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u968f\u673a\u79cd\u5b50<\/p>\n<\/td>\n | \n srand(2026) \u5728 testBPNN() \u4e2d\u53ea\u8c03\u7528\u4e00\u6b21<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u8f93\u51fa<\/p>\n<\/td>\n | \n \u63a7\u5236\u53f0+\u6587\u4ef6\u53cc\u8f93\u51fa，\u4fbf\u4e8e\u96c6\u6210\u5230\u8bc4\u4f30\u7cfb\u7edf<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u4e94\u3001MATLAB\u5b9e\u73b0<\/h2>\n%%
\n% @file : Alg_11_BPNN.m
\n% @brief : BP\u795e\u7ecf\u7f51\u7edc\u6a21\u578b
\n% @details : Backpropagation Neural Network，\u57fa\u4e8e\u8bef\u5dee\u53cd\u5411\u4f20\u64ad\u7684\u591a\u5c42\u524d\u9988
\n% \u76d1\u7763\u5b66\u4e60\u7f51\u7edc，\u7528\u4e8e\u975e\u7ebf\u6027\u62df\u5408\u4e0e\u6548\u80fd\u9884\u6d4b
\n% @author : \u4e09\u73af\u4e0a\u7684\u9a91\u58eb
\n% @date : 2026-05
\n% @version : v1.0
\n% @note : \u5df2\u56fa\u5b9a\u968f\u673a\u79cd\u5b50\u4fdd\u8bc1\u53ef\u590d\u73b0；\u9700\u76d1\u7763\u4fe1\u53f7(\u5982\u4e13\u5bb6\u7efc\u5408\u6548\u80fd\u503c)\u8bad\u7ec3
\n% @warning : \u968f\u673a\u521d\u59cb\u5316\u6613\u9677\u5c40\u90e8\u6781\u5c0f，\u7ed3\u679c\u5b58\u5728\u4e00\u5b9a\u6ce2\u52a8\u6027；\u9700\u9632\u8fc7\u62df\u5408
\nclc; clear; close all;<\/p>\n %% 1. \u6837\u672c\u6570\u636e（5\u4e2a\u88c5\u5907\u65b9\u6848，4\u8f93\u51651\u8f93\u51fa）
\n% \u8f93\u5165：\u63a2\u6d4b\u8ddd\u79bb \u53cd\u5e94\u65f6\u95f4 \u547d\u4e2d\u6982\u7387 \u6210\u672c（\u5df2\u6807\u51c6\u5316\u52300~1，\u8d8a\u5927\u8d8a\u597d）
\nP = [
\n 0.5294 0.1765 0.4118 1.0000; % \u65b9\u6848A
\n 0.2353 0.3529 0.1765 0.4118; % \u65b9\u6848B
\n 1.0000 0.0000 1.0000 0.0000; % \u65b9\u6848C
\n 0.0000 0.4706 0.0000 0.4118; % \u65b9\u6848D
\n 0.8235 0.6471 0.8235 0.7647 % \u65b9\u6848E
\n]'; % 4\u00d75，\u6bcf\u5217\u4e00\u4e2a\u6837\u672c<\/p>\n % \u8f93\u51fa：\u7efc\u5408\u6548\u80fd\u771f\u503c（\u7528TOPSIS\u5f97\u5206\u793a\u610f，0~1）
\nT = [0.6218 0.4873 0.7125 0.8031 0.6652]; % 1\u00d75<\/p>\n %% 2. \u7f51\u7edc\u7ed3\u6784
\ninNum = size(P,1); % 4
\nhidNum = 6;
\noutNum = size(T,1); % 1
\nS = size(P,2); % \u6837\u672c\u6570=5<\/p>\n %% 3. \u6743\u503c\u521d\u59cb\u5316（Xavier\u98ce\u683c，randn\u5728R2016a\u53ef\u7528）
\nrng(2026);
\nW1 = randn(hidNum, inNum) * sqrt(1\/inNum);
\nb1 = zeros(hidNum, 1);
\nW2 = randn(outNum, hidNum) * sqrt(1\/hidNum);
\nb2 = zeros(outNum, 1);<\/p>\n %% 4. \u8bad\u7ec3\u53c2\u6570
\nepochs = 6000;
\nlr = 0.05;
\nloss_rec = zeros(epochs,1);<\/p>\n sigmoid = @(x) 1.\/(1+exp(-x)); % \u533f\u540d\u51fd\u6570\u5728R2016a\u652f\u6301<\/p>\n %% 5. \u6279\u91cfBP\u8bad\u7ec3
\nfor epoch = 1:epochs
\n % —- \u524d\u5411 —-
\n Z1 = W1*P + repmat(b1, 1, S); % hidNum\u00d7S
\n A1 = sigmoid(Z1); % hidNum\u00d7S
\n Z2 = W2*A1 + repmat(b2, 1, S); % outNum\u00d7S
\n A2 = Z2; % purelin<\/p>\n % —- \u8bef\u5dee（MSE\/2） —-
\n E = A2 – T; % outNum\u00d7S
\n loss = mean(sum(E.^2,1))\/2;
\n loss_rec(epoch) = loss;<\/p>\n % —- \u53cd\u5411\u4f20\u64ad —-
\n dZ2 = E; % purelin\u5bfc\u6570=1
\n dW2 = dZ2 * A1' \/ S;
\n db2 = mean(dZ2, 2); % outNum\u00d71<\/p>\n dA1 = W2' * dZ2; % hidNum\u00d7S
\n dZ1 = dA1 .* A1 .* (1-A1); % sigmoid\u5bfc\u6570
\n dW1 = dZ1 * P' \/ S;
\n db1 = mean(dZ1, 2); % hidNum\u00d71<\/p>\n % —- \u66f4\u65b0 —-
\n W2 = W2 – lr*dW2;
\n b2 = b2 – lr*db2;
\n W1 = W1 – lr*dW1;
\n b1 = b1 – lr*db1;<\/p>\n if mod(epoch,500)==0
\n fprintf('Epoch %4d | MSE = %.6f\\\\n', epoch, loss);
\n end
\nend<\/p>\n %% 6. \u9884\u6d4b
\nZ1 = W1*P + repmat(b1,1,S);
\nA1 = sigmoid(Z1);
\nY_pred = W2*A1 + repmat(b2,1,S); % 1\u00d75<\/p>\n fprintf('\\\\n========== BP\u6548\u80fd\u8bc4\u4f30\u7ed3\u679c ==========\\\\n');
\nfprintf('%-8s %10s %10s %12s\\\\n','\u65b9\u6848','\u771f\u5b9e\u503c','\u9884\u6d4b\u503c','\u7edd\u5bf9\u8bef\u5dee');
\nlabels = {'A','B','C','D','E'};
\nfor i = 1:S
\n fprintf('\u65b9\u6848%s %10.4f %10.4f %12.4f\\\\n',…
\n labels{i}, T(i), Y_pred(i), abs(Y_pred(i)-T(i)));
\nend<\/p>\n %% 7. \u8bc4\u4ef7\u6307\u6807
\nMSE = mean((Y_pred – T).^2);
\nRMSE = sqrt(MSE);
\nR2 = 1 – sum((T-Y_pred).^2)\/sum((T-mean(T)).^2);
\nfprintf('\\\\nMSE=%.6f RMSE=%.4f R2=%.4f\\\\n', MSE, RMSE, R2);<\/p>\n %% 8. \u635f\u5931\u66f2\u7ebf
\nfigure('Color','w');
\nplot(loss_rec,'LineWidth',1.2); grid on;
\nxlabel('Epoch'); ylabel('MSE');
\ntitle('BP\u8bad\u7ec3\u635f\u5931\u66f2\u7ebf'); <\/p>\n \u516d\u3001C\u4e0eMATLAB\u5b9e\u73b0\u5bf9\u6bd4<\/h2>\n\n\n\u7ef4\u5ea6<\/p>\n MATLAB<\/p>\n C\u8bed\u8a00<\/p>\n<\/tr>\n \n\n| \n \u4ee3\u7801\u91cf<\/p>\n<\/td>\n | \n ~80\u884c<\/p>\n<\/td>\n | \n ~200\u884c<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u77e9\u9635\u8fd0\u7b97<\/p>\n<\/td>\n | \n \u539f\u751f\u652f\u6301，\u5411\u91cf\u5316<\/p>\n<\/td>\n | \n \u624b\u52a8\u5c55\u5f00\u5faa\u73af<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u521d\u59cb\u5316<\/p>\n<\/td>\n | \n randn \u6b63\u6001\u5206\u5e03<\/p>\n<\/td>\n | \n Xavier\u5747\u5300\u5206\u5e03<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u8fd0\u884c\u901f\u5ea6<\/p>\n<\/td>\n | \n \u8f83\u6162（\u89e3\u91ca\u6267\u884c）<\/p>\n<\/td>\n | \n \u5feb（\u7f16\u8bd1\u6267\u884c）<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u9002\u7528\u573a\u666f<\/p>\n<\/td>\n | \n \u7b97\u6cd5\u9a8c\u8bc1\u3001\u6559\u5b66<\/p>\n<\/td>\n | \n \u5de5\u7a0b\u5d4c\u5165\u3001\u5b9e\u65f6\u8bc4\u4f30<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u53ef\u79fb\u690d\u6027<\/p>\n<\/td>\n | \n \u4f9d\u8d56MATLAB\u73af\u5883<\/p>\n<\/td>\n | \n \u4efb\u610fC\u7f16\u8bd1\u5668<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \u4e24\u8005\u6570\u5b66\u7b49\u4ef7，\u5728\u76f8\u540c\u79cd\u5b50\u4e0b\u9884\u6d4b\u7ed3\u679c\u8bef\u5dee\u5728 $10^{-3}$ \u91cf\u7ea7（\u56e0\u521d\u59cb\u5316\u5206\u5e03\u4e0d\u540c）\u3002<\/p>\n \u4e03\u3001\u6a21\u578b\u8bc4\u4ef7\u4e0e\u6539\u8fdb\u65b9\u5411<\/h2>\n7.1 \u5f53\u524d\u6a21\u578b\u8868\u73b0<\/h3>\n\n- R\u00b2 > 0.95，\u62df\u5408\u4f18\u5ea6\u4f18\u79c0<\/li>\n
- RMSE < 0.05，\u6ee1\u8db3\u5de5\u7a0b\u7cbe\u5ea6\u9700\u6c42<\/li>\n
- 6000\u8f6e\u540e\u635f\u5931\u8d8b\u4e8e\u5e73\u7a33，\u6536\u655b\u826f\u597d<\/li>\n<\/ul>\n
7.2 \u5c40\u9650\u6027<\/h3>\n - \u6837\u672c\u91cf\u5c0f（5\u4e2a）：R\u00b2\u53ef\u80fd\u865a\u9ad8，\u6cdb\u5316\u80fd\u529b\u5b58\u7591<\/li>\n
- \u65e0\u9a8c\u8bc1\u96c6：\u65e0\u6cd5\u68c0\u6d4b\u8fc7\u62df\u5408<\/li>\n
- \u56fa\u5b9a\u5b66\u4e60\u7387：\u53ef\u80fd\u9677\u5165\u5c40\u90e8\u6781\u5c0f<\/li>\n
- \u65e0\u6b63\u5219\u5316：\u6743\u91cd\u53ef\u80fd\u8fc7\u5927<\/li>\n
7.3 \u6539\u8fdb\u5efa\u8bae<\/h3>\n\n\n\u6539\u8fdb\u65b9\u5411<\/p>\n \u65b9\u6cd5<\/p>\n<\/tr>\n \n\n| \n \u589e\u52a0\u6837\u672c<\/p>\n<\/td>\n | \n \u6269\u5145\u81f320+\u65b9\u6848，\u6216\u505aK\u6298\u4ea4\u53c9\u9a8c\u8bc1<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u9632\u8fc7\u62df\u5408<\/p>\n<\/td>\n | \n \u52a0\u5165L2\u6b63\u5219\u5316（\u6743\u91cd\u8870\u51cf）<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u4f18\u5316\u8bad\u7ec3<\/p>\n<\/td>\n | \n \u5b66\u4e60\u7387\u8870\u51cf \/ Adam\u4f18\u5316\u5668<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u7ed3\u6784\u4f18\u5316<\/p>\n<\/td>\n | \n \u4ea4\u53c9\u9a8c\u8bc1\u9009\u62e9\u9690\u5c42\u8282\u70b9\u6570<\/p>\n<\/td>\n<\/tr>\n | \n| \n \u65e9\u505c\u673a\u5236<\/p>\n<\/td>\n | \n \u9a8c\u8bc1\u96c6\u635f\u5931\u4e0d\u518d\u4e0b\u964d\u65f6\u505c\u6b62<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
\n\u516b\u3001\u603b\u7ed3<\/h2>\n\u672c\u6587\u7ed9\u51fa\u4e86BP\u795e\u7ecf\u7f51\u7edc\u4ece\u6570\u5b66\u539f\u7406 \u2192 MATLAB\u5feb\u901f\u9a8c\u8bc1 \u2192 C\u8bed\u8a00\u5de5\u7a0b\u5b9e\u73b0\u7684\u5b8c\u6574\u94fe\u8def\u3002\u6838\u5fc3\u8981\u70b9：<\/p>\n - \u524d\u5411\u4f20\u64ad + \u53cd\u5411\u4f20\u64ad + \u68af\u5ea6\u4e0b\u964d\u662fBP\u7684\u4e09\u5927\u652f\u67f1<\/li>\n
- Xavier\u521d\u59cb\u5316\u80fd\u6709\u6548\u6539\u5584\u6536\u655b<\/li>\n
- \u6279\u91cf\u8bad\u7ec3\u6bd4\u5728\u7ebf\u8bad\u7ec3\u66f4\u7a33\u5b9a<\/li>\n
- C\u548cMATLAB\u53cc\u5b9e\u73b0\u53ef\u4ee5\u4e92\u4e3a\u9a8c\u8bc1，\u517c\u987e\u5de5\u7a0b\u4e0e\u9a8c\u8bc1\u9700\u6c42<\/li>\n
\u4ee3\u7801\u5df2\u5f00\u6e90\u98ce\u683c\u5448\u73b0，\u53ef\u76f4\u63a5\u7f16\u8bd1\u8fd0\u884c\u3002\u5982\u9700\u6269\u5c55\u4e3a\u901a\u7528\u5e93\u6216\u52a0\u5165\u4ea4\u53c9\u9a8c\u8bc1，\u6b22\u8fce\u4ea4\u6d41\u8ba8\u8bba\u3002<\/p>\n
\n\u4e5d\u3001\u52d8\u8bef\u53ca\u66f4\u65b0\u8bf4\u660e<\/h2>\n\u672c\u6587\u5982\u6709\u758f\u6f0f\u6216\u8868\u8ff0\u4e0d\u5f53\u4e4b\u5904，\u6b22\u8fce\u5404\u4f4d\u8bfb\u8005\u5728\u8bc4\u8bba\u533a\u6307\u6b63，\u535a\u4e3b\u4f1a\u6301\u7eed\u5173\u6ce8\u53cd\u9988\u5e76\u53ca\u65f6\u4fee\u6b63\u4f18\u5316，\u529b\u6c42\u5185\u5bb9\u51c6\u786e\u53ef\u9760\u3002\u611f\u8c22\u5927\u5bb6\u7684\u76d1\u7763\u4e0e\u966a\u4f34\u3002<\/p>\n \u5982\u679c\u672c\u6587\u5bf9\u4f60\u7684\u7814\u7a76\u6216\u9879\u76ee\u6709\u5e2e\u52a9，\u6b22\u8fce\u70b9\u8d5e\u3001\u6536\u85cf\u3001\u5173\u6ce8\u4e09\u8fde！\u200b<\/p>\n \u7248\u6743\u58f0\u660e：\u672c\u6587\u4e3a\u539f\u521b\u6280\u672f\u6587\u7ae0，\u672a\u7ecf\u4f5c\u8005\u540c\u610f\u4e0d\u5f97\u8f6c\u8f7d\u3002\u4ee3\u7801\u9075\u5faaMIT\u5f00\u6e90\u534f\u8bae\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u4e00\u3001\u524d\u8a00
\n\u5728\u88c5\u5907\u8bba\u8bc1\u4e0e\u6548\u80fd\u8bc4\u4f30\u9886\u57df，\u5e38\u9762\u4e34\u8fd9\u6837\u4e00\u4e2a\u95ee\u9898：\u5df2\u77e5\u591a\u4e2a\u65b9\u6848\u7684\u82e5\u5e72\u6307\u6807\u53c2\u6570，\u5982\u4f55\u9884\u6d4b\u5176\u7efc\u5408\u6548\u80fd？
\n\u5f53\u6307\u6807\u4e0e\u6548\u80fd\u4e4b\u95f4\u5b58\u5728\u975e\u7ebf\u6027\u5173\u7cfb\u65f6，\u4f20\u7edf\u52a0\u6743\u6c42\u548c\u6216TOPSIS\u7b49\u65b9\u6cd5\u96be\u4ee5\u523b\u753b\u590d\u6742\u7684\u6620\u5c04\u89c4\u5f8b\u3002\u6b64\u65f6，BP（Backpropagation）\u795e\u7ecf\u7f51\u7edc\u4f5c\u4e3a\u4e00\u79cd\u7ecf\u5178\u7684\u76d1\u7763\u5b66\u4e60\u6a21\u578b，\u80fd\u591f\u901a\u8fc7\u6837\u672c\u5b66\u4e60\u8f93\u5165\u4e0e\u8f93\u51fa\u4e4b\u95f4\u7684\u975e\u7ebf\u6027\u6620\u5c04，\u662f\u5de5\u7a0b\u5b9e\u8df5\u4e2d\u5e38\u7528<\/p>\n","protected":false},"author":2,"featured_media":103112,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[587,11758,11759,55,1284],"topic":[],"class_list":["post-103113","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-server","tag-587","tag-bpnn","tag-11759","tag-c","tag-matlab"],"yoast_head":"\n \u3010\u6548\u80fd\u8bc4\u4f30\u7cfb\u521707\u3011BP\u795e\u7ecf\u7f51\u7edc\uff1aC\u4e0eMATLAB\u53cc\u5b9e\u73b0\u53ca\u6548\u80fd\u8bc4\u4f30\uff08\u9644\u6e90\u7801\uff09 - 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