Page 194 - 《软件学报》2025年第9期

胡思宇 等: 基于层重组扩展卡尔曼滤波的神经网络力场训练 4105

表 5 多模型在 Adam 和层重组卡尔曼滤波优化器下的收敛误差

体系 模型/方法 训练集 RMSE E 训练集 RMSE F 测试集 RMSE E 测试集 RMSE F
(Adam/RLEKF) (Adam/RLEKF) (Adam/RLEKF) (Adam/KF)
2B3B-Cosine 0.054 4/0.039 2 0.061 5/0.057 7 0.060 3/0.048 7 0.073 3/0.069 8
2B3B-Gaussian 0.079 5/0.048 0 0.070 2/0.061 0 0.084 5/0.058 5 0.081 5/0.069 9
Cu1646
MTP 0.072 4/0.049 9 0.073 2/0.070 2 0.072 8/0.064 9 0.089 0/0.089 3
SNAP 0.086 7/0.063 1 0.118/0.092 6 0.125/0.085 6 0.157/0.122
2B3B-Cosine 0.004 92/0.003 42 0.013 1/0.010 0 0.008 35/0.008 40 0.016 4/0.019 6
2B3B-Gaussian 0.006 14/0.004 72 0.012 5/0.011 4 0.007 74/0.007 63 0.013 8/0.014 2
Ag2015
MTP 0.006 33/0.003 44 0.014 4/0.008 49 0.011 9/0.008 33 0.017 8/0.016 5
SNAP 0.008 99/0.004 88 0.020 7/0.012 4 0.013 9/0.012 0 0.025 3/0.021 8
2B3B-Cosine 0.040 0/0.029 3 0.061 4/0.055 5 0.040 6/0.038 6 0.061 5/0.061 5
2B3B-Gaussian 0.056 6/0.038 1 0.073 5/0.064 5 0.054 2/0.043 2 0.072 1/0.066 2
Al4000
MTP 0.038 5/0.024 7 0.058 8/0.048 1 0.034 2/0.031 2 0.058 4/0.054 8
SNAP 0.051 9/0.032 2 0.080 6/0.066 8 0.051 3/0.035 2 0.083 6/0.077 3
2B3B-Cosine 0.021 5/0.014 9 0.042 5/0.035 4 0.016 9/0.018 7 0.049 8/0.047 8
2B3B-Gaussian 0.026 7/0.012 8 0.070 9/0.034 3 0.025 1/0.022 2 0.075 9/0.043 4
C4000
MTP 0.131/0.069 9 0.330/0.200 0.129/0.171 0.354/0.283
SNAP 0.222/0.129 0.562/0.481 0.365/0.233 0.732/0.702
2B3B-Cosine 0.152/0.012 4 0.026 3/0.026 1 0.290/0.174 0.027 4/0.028 4
2B3B-Gaussian 0.091 7/0.017 0 0.020 0/0.020 1 0.178/0.123 0.019 7/0.020 4
Li1000
MTP 0.031 4/0.003 40 0.009 22/0.008 88 0.039 7/0.043 1 0.009 52/0.009 43
SNAP 0.095 8/0.006 67 0.020 2/0.016 1 0.113/0.078 8 0.021 5/0.017 4
2B3B-Cosine 0.012 2/0.008 79 0.020 9/0.017 9 0.008 73/0.010 9 0.022 6/0.029 6
2B3B-Gaussian 0.015 8/0.009 21 0.023 0/0.019 7 0.014 6/0.013 5 0.023 4/0.023 7
Mg4000
MTP 0.007 18/0.002 86 0.014 1/0.010 5 0.006 52/0.006 25 0.015 8/0.015 3
SNAP 0.012 5/0.002 87 0.018 3/0.011 4 0.012 0/0.007 85 0.021 3/0.015 9
2B3B-Cosine 0.198/0.020 5 0.080 9/0.073 1 0.297/0.249 0.135/0.145
2B3B-Gaussian 0.249/0.052 1 0.106/0.087 5 0.462/0.217 0.150/0.139
S2000
MTP 0.202/0.022 5 0.147/0.089 3 0.438/0.322 0.213/0.195
SNAP 0.499/0.056 9 0.266/0.198 0.661/0.511 0.351/0.328
2B3B-Cosine 0.047 9/0.038 4 0.051 9/0.047 3 0.037 7/0.032 0 0.053 5/0.051 8
2B3B-Gaussian 0.067 9/0.041 7 0.059 1/0.045 2 0.051 1/0.046 9 0.060 8/0.047 8
Si3000
MTP 0.044 8/0.017 8 0.062 2/0.034 5 0.031 7/0.023 4 0.063 9/0.038 9
SNAP 0.052 9/0.025 9 0.075 3/0.045 5 0.042 2/0.023 9 0.079 3/0.049 2
2B3B-Cosine 0.023 6/0.009 00 0.048 8/0.039 4 0.033 2/0.244 0.086 4/1.10
2B3B-Gaussian 0.041 6/0.267 0.071 1/3.03 0.030 8/0.024 0.081 5/0.067 6
H 2 O4000
MTP 0.030 7/0.010 1 0.049 6/0.134 0.099 6/0.035 5 0.129/0.314
SNAP 0.204/0.226 0.575/0.748 0.313/0.339 0.690/0.791
2B3B-Cosine 0.110/0.056 6 0.178/0.174 0.112/0.192 0.188/0.204
Cu+C2000 MTP 0.793/0.098 1 0.190/0.190 0.724/0.305 0.209/0.725
SNAP 0.631/0.167 0.469/0.507 0.467/0.350 0.495/0.546
2B3B-Cosine 0.006 79/0.003 20 0.008 87/0.007 38 0.613/1.38 4.15/2.73
2B3B-Gaussian 0.006 90/0.003 38 0.007 85/0.006 59 0.008 28/0.003 82 0.008 54/0.007 34
NaCl3193
MTP 0.004 85/0.001 71 0.007 38/0.005 71 0.004 74/0.004 03 0.008 58/0.006 71
SNAP 0.013 2/0.005 91 0.017 6/0.012 3 0.015 0/0.010 8 0.018 6/0.014 8

3.2 收敛速度
我们考虑 8 个单元素体系的数据集的收敛速度, 鉴于表 5 中单元素体系在一阶 Adam 优化器和二阶层重组卡
尔曼滤波优化器的收敛精度相当, 因此比较 8 个单元素体系收敛到表 5 所示的精度时的收敛速度. 图 6(a)–(d) 分
别表示在 8 个单元素的体系, 4 种不同模型下, Adam 优化器训练 1 000 个 epoch, 层重组卡尔曼滤波优化器训练