├── notes ├── notes.out ├── s0.png ├── error.png ├── notes.pdf ├── s1000.png ├── s500.png ├── s5000.png ├── MscaleNet2.png ├── notes.synctex.gz ├── notes.aux ├── notes.tex └── notes.log ├── exp ├── u0.png ├── u500.png ├── error.png ├── u1000.png ├── u5000.png ├── sin_M │ └── 62178 │ │ ├── data.pkl │ │ ├── error.png │ │ ├── loss.png │ │ ├── loss_bd.png │ │ ├── u_epoch_0.png │ │ ├── u_epoch_1000.png │ │ ├── u_epoch_1500.png │ │ ├── u_epoch_2000.png │ │ ├── u_epoch_2500.png │ │ ├── u_epoch_3000.png │ │ ├── u_epoch_3500.png │ │ ├── u_epoch_4000.png │ │ ├── u_epoch_4500.png │ │ ├── u_epoch_500.png │ │ ├── u_epoch_5000.png │ │ └── code.py └── sin_N │ └── 80509 │ ├── data.pkl │ ├── error.png │ ├── loss.png │ ├── loss_bd.png │ ├── u_epoch_0.png │ ├── u_epoch_1000.png │ ├── u_epoch_1500.png │ ├── u_epoch_2000.png │ ├── u_epoch_2500.png │ ├── u_epoch_3000.png │ ├── u_epoch_3500.png │ ├── u_epoch_4000.png │ ├── u_epoch_4500.png │ ├── u_epoch_500.png │ ├── u_epoch_5000.png │ └── code.py ├── code 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-------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # mscalednn 2 | This is an example code of solving Poisson Equation by MscaleDNN. The detail of MscaleDNN can be found in paper 3 | https://arxiv.org/abs/2007.11207, CiCP 4 | The underlying idea comes from frequency principle of deep learning, see https://arxiv.org/abs/1901.06523 5 | -------------------------------------------------------------------------------- /notes/notes.aux: -------------------------------------------------------------------------------- 1 | \relax 2 | \providecommand\hyper@newdestlabel[2]{} 3 | \providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} 4 | \HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined 5 | \global\let\oldcontentsline\contentsline 6 | \gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} 7 | \global\let\oldnewlabel\newlabel 8 | \gdef\newlabel#1#2{\newlabelxx{#1}#2} 9 | \gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} 10 | \AtEndDocument{\ifx\hyper@anchor\@undefined 11 | \let\contentsline\oldcontentsline 12 | \let\newlabel\oldnewlabel 13 | \fi} 14 | \fi} 15 | \global\let\hyper@last\relax 16 | \gdef\HyperFirstAtBeginDocument#1{#1} 17 | \providecommand*\HyPL@Entry[1]{} 18 | \citation{deep-ritz} 19 | \HyPL@Entry{0<>} 20 | \citation{mscale} 21 | \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Mscale网络结构\relax }}{3}{figure.caption.4}\protected@file@percent } 22 | \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} 23 | \newlabel{net}{{1}{3}{Mscale网络结构\relax }{figure.caption.4}{}} 24 | \bibcite{deep-ritz}{1} 25 | \bibcite{mscale}{2} 26 | \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces 求解过程\relax }}{12}{figure.caption.15}\protected@file@percent } 27 | \newlabel{proc}{{2}{12}{求解过程\relax }{figure.caption.15}{}} 28 | \@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {0步}}}{12}{subfigure.2.1}\protected@file@percent } 29 | \@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {500步}}}{12}{subfigure.2.2}\protected@file@percent } 30 | \@writefile{lof}{\contentsline {subfigure}{\numberline{(c)}{\ignorespaces {1000步}}}{12}{subfigure.2.3}\protected@file@percent } 31 | \@writefile{lof}{\contentsline {subfigure}{\numberline{(d)}{\ignorespaces {5000步}}}{12}{subfigure.2.4}\protected@file@percent } 32 | \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces 误差下降曲线\relax }}{12}{figure.caption.16}\protected@file@percent } 33 | \newlabel{error}{{3}{12}{误差下降曲线\relax }{figure.caption.16}{}} 34 | -------------------------------------------------------------------------------- /code/plot_result.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import os 8 | import pickle 9 | import numpy as np 10 | import matplotlib.pyplot as plt 11 | from scipy import signal 12 | 13 | import matplotlib 14 | matplotlib.use('Agg') 15 | Leftp = 0.18 16 | Bottomp = 0.18 17 | Widthp = 0.88 - Leftp 18 | Heightp = 0.9 - Bottomp 19 | pos = [Leftp, Bottomp, Widthp, Heightp] 20 | 21 | def save_fig(pltm, fntmp,fp=0,ax=0,isax=0,iseps=0,isShowPic=0):# Save the figure 22 | if isax==1: 23 | pltm.rc('xtick',labelsize=18) 24 | pltm.rc('ytick',labelsize=18) 25 | ax.set_position(pos, which='both') 26 | fnm = '%s.png'%(fntmp) 27 | pltm.savefig(fnm) 28 | if iseps: 29 | fnm = '%s.eps'%(fntmp) 30 | pltm.savefig(fnm, format='eps', dpi=600) 31 | if fp!=0: 32 | fp.savefig("%s.pdf"%(fntmp), bbox_inches='tight') 33 | if isShowPic==1: 34 | pltm.show() 35 | elif isShowPic==-1: 36 | return 37 | else: 38 | pltm.close() 39 | 40 | 41 | with open(os.path.join("..", "exp", "sin_N", "80509", "data.pkl"), "rb") as f: 42 | R1 = pickle.load(f) 43 | with open(os.path.join("..", "exp", "sin_M", "62178", "data.pkl"), "rb") as f: 44 | R2 = pickle.load(f) 45 | 46 | 47 | record_path=os.path.join("..", "exp") 48 | plt.figure() 49 | ax = plt.gca() 50 | plt.semilogy(R1["error"], label="Normal") 51 | plt.semilogy(R2["error"], label="Mscale") 52 | plt.legend(fontsize=14) 53 | fntmp = os.path.join(record_path, 'error') 54 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 55 | 56 | for k in [0, 500, 1000, 5000]: 57 | plt.figure() 58 | plt.plot(R2["x_samp"], R2["u_samp_"+str(k)], 'r--',label='dnn') 59 | plt.plot(R2["x_samp"], R2["u_samp_true"], 'b-',label='true') 60 | plt.legend(fontsize=14) 61 | plt.title('epoch '+str(k),fontsize=14) 62 | fntmp = os.path.join(record_path, 'u'+str(k)) 63 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 64 | -------------------------------------------------------------------------------- /code/my_act.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import tensorflow as tf 8 | import numpy as np 9 | 10 | 11 | def relu2(x): 12 | return tf.nn.relu(x) ** 2 13 | 14 | 15 | def relu3(x): 16 | return tf.nn.relu(x) ** 3 17 | 18 | 19 | def srelu(x): 20 | return tf.nn.relu(1 - x) * tf.nn.relu(x) 21 | 22 | 23 | def wave(x): 24 | return ( 25 | tf.nn.relu(x) 26 | - 2 * tf.nn.relu(x - 1 / 4) 27 | + 2 * tf.nn.relu(x - 3 / 4) 28 | - tf.nn.relu(x - 1) 29 | ) 30 | 31 | 32 | def ex2(x): 33 | return tf.exp(-tf.square(x)) 34 | 35 | 36 | def xex2(x): 37 | return x * tf.exp(-tf.square(x)) 38 | 39 | 40 | def phi2(x): 41 | return tf.nn.relu(x) - 2 * tf.nn.relu(x - 1) + tf.nn.relu(x - 2) 42 | 43 | 44 | def psi2(x): 45 | return ( 46 | 1 / 12 * tf.nn.relu(2 * x) 47 | - 2 / 3 * tf.nn.relu(2 * x - 1) 48 | + 23 / 12 * tf.nn.relu(2 * x - 2) 49 | - 8 / 3 * tf.nn.relu(2 * x - 3) 50 | + 23 / 12 * tf.nn.relu(2 * x - 4) 51 | - 2 / 3 * tf.nn.relu(2 * x - 5) 52 | + 1 / 12 * tf.nn.relu(2 * x - 6) 53 | ) 54 | 55 | 56 | def phi3(x): 57 | return ( 58 | 1 / 2 * tf.nn.relu(x - 0) ** 2 59 | - 3 / 2 * tf.nn.relu(x - 1) ** 2 60 | + 3 / 2 * tf.nn.relu(x - 2) ** 2 61 | - 1 / 2 * tf.nn.relu(x - 3) ** 2 62 | ) 63 | 64 | 65 | def psi3(x): 66 | return ( 67 | 1 / 960 * tf.nn.relu(2 * x - 0) ** 2 68 | - 32 / 960 * tf.nn.relu(2 * x - 1) ** 2 69 | + 237 / 960 * tf.nn.relu(2 * x - 2) ** 2 70 | - 832 / 960 * tf.nn.relu(2 * x - 3) ** 2 71 | + 1682 / 960 * tf.nn.relu(2 * x - 4) ** 2 72 | - 2112 / 960 * tf.nn.relu(2 * x - 5) ** 2 73 | + 1682 / 960 * tf.nn.relu(2 * x - 6) ** 2 74 | - 832 / 960 * tf.nn.relu(2 * x - 7) ** 2 75 | + 237 / 960 * tf.nn.relu(2 * x - 8) ** 2 76 | - 32 / 960 * tf.nn.relu(2 * x - 9) ** 2 77 | + 1 / 960 * tf.nn.relu(2 * x - 10) ** 2 78 | ) 79 | 80 | def s2relu(x): 81 | return ( 82 | tf.sin(2*np.pi*x)*srelu(x) 83 | ) 84 | 85 | 86 | act_dict = { 87 | "relu": tf.nn.relu, 88 | "relu2": relu2, 89 | "relu3": relu3, 90 | "sigmoid": tf.nn.sigmoid, 91 | "tanh": tf.nn.tanh, 92 | "softplus": tf.nn.softplus, 93 | "sin": tf.sin, 94 | "cos": tf.cos, 95 | "ex2": ex2, 96 | "xex2": xex2, 97 | "srelu": srelu, 98 | "wave": wave, 99 | "phi2": phi2, 100 | "psi2": psi2, 101 | "phi3": phi3, 102 | "psi3": psi3, 103 | "s2relu": s2relu, 104 | } 105 | -------------------------------------------------------------------------------- /code/ritzN.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import os 8 | import time 9 | import shutil 10 | import pickle 11 | 12 | import tensorflow as tf 13 | import numpy as np 14 | import matplotlib.pyplot as plt 15 | 16 | from my_act import act_dict 17 | 18 | import matplotlib 19 | matplotlib.use('Agg') 20 | Leftp = 0.18 21 | Bottomp = 0.18 22 | Widthp = 0.88 - Leftp 23 | Heightp = 0.9 - Bottomp 24 | pos = [Leftp, Bottomp, Widthp, Heightp] 25 | 26 | def save_fig(pltm, fntmp,fp=0,ax=0,isax=0,iseps=0,isShowPic=0):# Save the figure 27 | if isax==1: 28 | pltm.rc('xtick',labelsize=18) 29 | pltm.rc('ytick',labelsize=18) 30 | ax.set_position(pos, which='both') 31 | fnm = '%s.png'%(fntmp) 32 | pltm.savefig(fnm) 33 | if iseps: 34 | fnm = '%s.eps'%(fntmp) 35 | pltm.savefig(fnm, format='eps', dpi=600) 36 | if fp!=0: 37 | fp.savefig("%s.pdf"%(fntmp), bbox_inches='tight') 38 | if isShowPic==1: 39 | pltm.show() 40 | elif isShowPic==-1: 41 | return 42 | else: 43 | pltm.close() 44 | 45 | 46 | # %% parameters 47 | R = {} 48 | 49 | R["seed"] = 0 50 | 51 | R["size_it"] = 1000 52 | R["size_bd"] = 100 53 | 54 | R["penalty"] = 1e3 55 | 56 | R["act_name"] = ("sin",) * 3 57 | R["hidden_units"] = (180,) * 3 58 | R["learning_rate"] = 1e-4 59 | R["lr_decay"] = 5e-7 60 | 61 | R["varcoe"] = 0.5 62 | 63 | R["total_step"] = 5000 64 | R["resamp_epoch"] = 1 65 | R["plot_epoch"] = 500 66 | 67 | R["record_path"] = os.path.join("..", "exp", "sin_N",str(np.random.randint(10000,99999))) 68 | 69 | # %% generate data in domain 70 | 71 | R["dimension"] = dim = 1 72 | R["area_it"] = 2 73 | R["area_bd"] = 1 74 | 75 | # interior data 76 | def rand_it(size): 77 | x_it = np.random.rand(size, dim) * 2 - 1 78 | return x_it.astype(np.float32) 79 | 80 | 81 | # boundary data 82 | def rand_bd(size): 83 | 84 | x_bd = np.random.rand(size, dim) * 2 - 1 85 | ind = np.random.randint(dim, size=size) 86 | x01 = np.random.randint(2, size=size) * 2 - 1 87 | for ii in range(size): 88 | x_bd[ii, ind[ii]] = x01[ii] 89 | 90 | return x_bd.astype(np.float32) 91 | 92 | 93 | # %% PDE problem 94 | # - (a(x) u'(x))' + c(x) u(x) = f(x) 95 | R["mu"] = mu = 6 * np.pi 96 | 97 | 98 | def u(x): 99 | u = np.sin(mu * x) 100 | return u.astype(np.float32).reshape((-1, 1)) 101 | 102 | 103 | def a(x): 104 | a = np.ones((x.shape[0], 1)) 105 | return a.astype(np.float32).reshape((-1, 1)) 106 | 107 | 108 | def c(x): 109 | c = np.zeros((x.shape[0], 1)) 110 | return c.astype(np.float32).reshape((-1, 1)) 111 | 112 | 113 | def f(x): 114 | f = mu * mu * np.sin(mu * x) 115 | return f.astype(np.float32).reshape((-1, 1)) 116 | 117 | 118 | # %% save parameters 119 | 120 | # prepare folder 121 | if not os.path.isdir(R["record_path"]): 122 | os.mkdir(R["record_path"]) 123 | 124 | # save current code 125 | save_file = os.path.join(R["record_path"], "code.py") 126 | shutil.copyfile(__file__, save_file) 127 | 128 | # %% set seed 129 | tf.set_random_seed(R["seed"]) 130 | tf.random.set_random_seed(R["seed"]) 131 | np.random.seed(R["seed"]) 132 | 133 | # %% get test points 134 | x_test = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 135 | 136 | R["x_test"] = x_test.astype(np.float32) 137 | R["u_test_true"] = u(R["x_test"]) 138 | 139 | # %% get sample points 140 | x_samp = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 141 | 142 | R["x_samp"] = x_samp.astype(np.float32) 143 | R["u_samp_true"] = u(R["x_samp"]) 144 | 145 | # %% normal neural network 146 | units = (dim,) + R["hidden_units"] + (1,) 147 | 148 | 149 | def neural_net(x): 150 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 151 | 152 | y = x 153 | 154 | for i in range(len(units) - 2): 155 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 156 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 157 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 158 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 159 | 160 | W = tf.get_variable(name="W" + str(i), initializer=init_W) 161 | b = tf.get_variable(name="b" + str(i), initializer=init_b) 162 | 163 | y = act_dict[R["act_name"][i]](tf.matmul(y, W) + b) 164 | 165 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 166 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 167 | init_b = np.random.randn(units[-1]).astype(np.float32) 168 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 169 | 170 | W = tf.get_variable(name="W" + str(len(units) - 1), initializer=init_W) 171 | b = tf.get_variable(name="b" + str(len(units) - 1), initializer=init_b) 172 | 173 | y = tf.matmul(y, W) + b 174 | 175 | return y 176 | 177 | 178 | # %% loss and optimizer ("V" for variable) 179 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 180 | 181 | Vx_it = tf.placeholder(tf.float32, shape=(None, dim)) 182 | Vx_bd = tf.placeholder(tf.float32, shape=(None, dim)) 183 | 184 | Va_it = tf.placeholder(tf.float32, shape=(None, 1)) 185 | Vc_it = tf.placeholder(tf.float32, shape=(None, 1)) 186 | Vf_it = tf.placeholder(tf.float32, shape=(None, 1)) 187 | 188 | Vu_true_bd = tf.placeholder(tf.float32, shape=(None, 1)) 189 | 190 | Vu_it = neural_net(Vx_it) 191 | Vu_bd = neural_net(Vx_bd) 192 | 193 | Vdu_it = tf.gradients(Vu_it, Vx_it)[0] 194 | 195 | Vloss_it = R["area_it"] * tf.reduce_mean( 196 | 1 / 2 * Va_it * tf.reduce_sum(tf.square(Vdu_it), axis=1, keepdims=True) 197 | + 1 / 2 * Vc_it * tf.square(Vu_it) 198 | - Vf_it * Vu_it 199 | ) 200 | Vloss_bd = R["area_bd"] * tf.reduce_mean(tf.square(Vu_bd - Vu_true_bd)) 201 | 202 | Vloss = Vloss_it + R["penalty"] * Vloss_bd 203 | 204 | learning_rate = tf.placeholder_with_default(input=1e-3, shape=[], name="lr") 205 | optimizer = tf.train.AdamOptimizer(learning_rate) 206 | train_op = optimizer.minimize(Vloss) 207 | 208 | # %% train model 209 | t0 = time.time() 210 | loss, loss_bd = [], [] 211 | error = [] 212 | lr = 1 * R["learning_rate"] 213 | 214 | config = tf.ConfigProto() 215 | config.gpu_options.allow_growth = True 216 | with tf.Session(config=config) as sess: 217 | 218 | sess.run(tf.global_variables_initializer()) 219 | 220 | for epoch in range(R["total_step"] + 1): 221 | 222 | # generate new data ("N" for number) 223 | if epoch % R["resamp_epoch"] == 0: 224 | 225 | Nx_it = rand_it(R["size_it"]) 226 | Nx_bd = rand_bd(R["size_bd"]) 227 | 228 | Na_it = a(Nx_it) 229 | Nc_it = c(Nx_it) 230 | Nf_it = f(Nx_it) 231 | 232 | Nu_bd = u(Nx_bd) 233 | 234 | # train neural network 235 | lr = (1 - R["lr_decay"]) * lr 236 | 237 | _, Nloss, Nloss_bd = sess.run( 238 | [train_op, Vloss, Vloss_bd], 239 | feed_dict={ 240 | Vx_it: Nx_it, 241 | Vx_bd: Nx_bd, 242 | Va_it: Na_it, 243 | Vc_it: Nc_it, 244 | Vf_it: Nf_it, 245 | Vu_true_bd: Nu_bd, 246 | learning_rate: lr, 247 | }, 248 | ) 249 | 250 | # get test error 251 | R["u_test"] = sess.run(Vu_it, feed_dict={Vx_it: R["x_test"]}) 252 | Nerror = R["area_it"] * np.mean((R["u_test"] - R["u_test_true"]) ** 2) 253 | 254 | loss.append(Nloss) 255 | loss_bd.append(Nloss_bd) 256 | error.append(Nerror) 257 | 258 | # show current state 259 | if epoch % R["plot_epoch"] == 0: 260 | 261 | print("epoch %d, time %.3f" % (epoch, time.time() - t0)) 262 | print("total loss %f, boundary loss %f" % (Nloss, Nloss_bd)) 263 | print("interior error %f" % (Nerror)) 264 | 265 | R["time"] = time.time() - t0 266 | R["loss"] = np.array(loss) 267 | R["loss_bd"] = np.array(loss_bd) 268 | R["error"] = np.array(error) 269 | 270 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 271 | R["u_samp_" + str(epoch)] = u_samp 272 | 273 | plt.figure() 274 | ax = plt.gca() 275 | plt.semilogy(R["error"]) 276 | plt.title('error',fontsize=15) 277 | fntmp = os.path.join(R["record_path"], 'error') 278 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 279 | 280 | plt.figure() 281 | ax = plt.gca() 282 | plt.plot(R["loss"]) 283 | plt.title('loss',fontsize=15) 284 | fntmp = os.path.join(R["record_path"], 'loss') 285 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 286 | 287 | plt.figure() 288 | ax = plt.gca() 289 | plt.semilogy(R["loss_bd"]) 290 | plt.title('loss_bd',fontsize=15) 291 | fntmp = os.path.join(R["record_path"], 'loss_bd') 292 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 293 | 294 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 295 | R["u_samp_" + str(epoch)] = u_samp 296 | 297 | if R['dimension']==1: 298 | plt.figure() 299 | ax = plt.gca() 300 | plt.plot(R["x_samp"], R["u_samp_"+str(epoch)], 'r--',label='dnn') 301 | plt.plot(R["x_samp"], R["u_samp_true"], 'b-',label='true') 302 | plt.title('u',fontsize=15) 303 | plt.legend(fontsize=18) 304 | fntmp = os.path.join(R["record_path"], 'u_epoch_%s'%(epoch)) 305 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 306 | 307 | 308 | # %% save data 309 | data_dir = os.path.join(R["record_path"], "data.pkl") 310 | with open(data_dir, "wb") as file: 311 | pickle.dump(R, file) 312 | 313 | # %% save data 314 | data_dir = os.path.join(R["record_path"], "data.pkl") 315 | with open(data_dir, "wb") as file: 316 | pickle.dump(R, file) 317 | -------------------------------------------------------------------------------- /exp/sin_N/80509/code.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import os 8 | import time 9 | import shutil 10 | import pickle 11 | 12 | import tensorflow as tf 13 | import numpy as np 14 | import matplotlib.pyplot as plt 15 | 16 | from my_act import act_dict 17 | 18 | import matplotlib 19 | matplotlib.use('Agg') 20 | Leftp = 0.18 21 | Bottomp = 0.18 22 | Widthp = 0.88 - Leftp 23 | Heightp = 0.9 - Bottomp 24 | pos = [Leftp, Bottomp, Widthp, Heightp] 25 | 26 | def save_fig(pltm, fntmp,fp=0,ax=0,isax=0,iseps=0,isShowPic=0):# Save the figure 27 | if isax==1: 28 | pltm.rc('xtick',labelsize=18) 29 | pltm.rc('ytick',labelsize=18) 30 | ax.set_position(pos, which='both') 31 | fnm = '%s.png'%(fntmp) 32 | pltm.savefig(fnm) 33 | if iseps: 34 | fnm = '%s.eps'%(fntmp) 35 | pltm.savefig(fnm, format='eps', dpi=600) 36 | if fp!=0: 37 | fp.savefig("%s.pdf"%(fntmp), bbox_inches='tight') 38 | if isShowPic==1: 39 | pltm.show() 40 | elif isShowPic==-1: 41 | return 42 | else: 43 | pltm.close() 44 | 45 | 46 | # %% parameters 47 | R = {} 48 | 49 | R["seed"] = 0 50 | 51 | R["size_it"] = 1000 52 | R["size_bd"] = 100 53 | 54 | R["penalty"] = 1e3 55 | 56 | R["act_name"] = ("sin",) * 3 57 | R["hidden_units"] = (180,) * 3 58 | R["learning_rate"] = 1e-4 59 | R["lr_decay"] = 5e-7 60 | 61 | R["varcoe"] = 0.5 62 | 63 | R["total_step"] = 5000 64 | R["resamp_epoch"] = 1 65 | R["plot_epoch"] = 500 66 | 67 | R["record_path"] = os.path.join("..", "exp", "sin_N",str(np.random.randint(10000,99999))) 68 | 69 | # %% generate data in domain 70 | 71 | R["dimension"] = dim = 1 72 | R["area_it"] = 2 73 | R["area_bd"] = 1 74 | 75 | # interior data 76 | def rand_it(size): 77 | x_it = np.random.rand(size, dim) * 2 - 1 78 | return x_it.astype(np.float32) 79 | 80 | 81 | # boundary data 82 | def rand_bd(size): 83 | 84 | x_bd = np.random.rand(size, dim) * 2 - 1 85 | ind = np.random.randint(dim, size=size) 86 | x01 = np.random.randint(2, size=size) * 2 - 1 87 | for ii in range(size): 88 | x_bd[ii, ind[ii]] = x01[ii] 89 | 90 | return x_bd.astype(np.float32) 91 | 92 | 93 | # %% PDE problem 94 | # - (a(x) u'(x))' + c(x) u(x) = f(x) 95 | R["mu"] = mu = 6 * np.pi 96 | 97 | 98 | def u(x): 99 | u = np.sin(mu * x) 100 | return u.astype(np.float32).reshape((-1, 1)) 101 | 102 | 103 | def a(x): 104 | a = np.ones((x.shape[0], 1)) 105 | return a.astype(np.float32).reshape((-1, 1)) 106 | 107 | 108 | def c(x): 109 | c = np.zeros((x.shape[0], 1)) 110 | return c.astype(np.float32).reshape((-1, 1)) 111 | 112 | 113 | def f(x): 114 | f = mu * mu * np.sin(mu * x) 115 | return f.astype(np.float32).reshape((-1, 1)) 116 | 117 | 118 | # %% save parameters 119 | 120 | # prepare folder 121 | if not os.path.isdir(R["record_path"]): 122 | os.mkdir(R["record_path"]) 123 | 124 | # save current code 125 | save_file = os.path.join(R["record_path"], "code.py") 126 | shutil.copyfile(__file__, save_file) 127 | 128 | # %% set seed 129 | tf.set_random_seed(R["seed"]) 130 | tf.random.set_random_seed(R["seed"]) 131 | np.random.seed(R["seed"]) 132 | 133 | # %% get test points 134 | x_test = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 135 | 136 | R["x_test"] = x_test.astype(np.float32) 137 | R["u_test_true"] = u(R["x_test"]) 138 | 139 | # %% get sample points 140 | x_samp = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 141 | 142 | R["x_samp"] = x_samp.astype(np.float32) 143 | R["u_samp_true"] = u(R["x_samp"]) 144 | 145 | # %% normal neural network 146 | units = (dim,) + R["hidden_units"] + (1,) 147 | 148 | 149 | def neural_net(x): 150 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 151 | 152 | y = x 153 | 154 | for i in range(len(units) - 2): 155 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 156 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 157 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 158 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 159 | 160 | W = tf.get_variable(name="W" + str(i), initializer=init_W) 161 | b = tf.get_variable(name="b" + str(i), initializer=init_b) 162 | 163 | y = act_dict[R["act_name"][i]](tf.matmul(y, W) + b) 164 | 165 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 166 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 167 | init_b = np.random.randn(units[-1]).astype(np.float32) 168 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 169 | 170 | W = tf.get_variable(name="W" + str(len(units) - 1), initializer=init_W) 171 | b = tf.get_variable(name="b" + str(len(units) - 1), initializer=init_b) 172 | 173 | y = tf.matmul(y, W) + b 174 | 175 | return y 176 | 177 | 178 | # %% loss and optimizer ("V" for variable) 179 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 180 | 181 | Vx_it = tf.placeholder(tf.float32, shape=(None, dim)) 182 | Vx_bd = tf.placeholder(tf.float32, shape=(None, dim)) 183 | 184 | Va_it = tf.placeholder(tf.float32, shape=(None, 1)) 185 | Vc_it = tf.placeholder(tf.float32, shape=(None, 1)) 186 | Vf_it = tf.placeholder(tf.float32, shape=(None, 1)) 187 | 188 | Vu_true_bd = tf.placeholder(tf.float32, shape=(None, 1)) 189 | 190 | Vu_it = neural_net(Vx_it) 191 | Vu_bd = neural_net(Vx_bd) 192 | 193 | Vdu_it = tf.gradients(Vu_it, Vx_it)[0] 194 | 195 | Vloss_it = R["area_it"] * tf.reduce_mean( 196 | 1 / 2 * Va_it * tf.reduce_sum(tf.square(Vdu_it), axis=1, keepdims=True) 197 | + 1 / 2 * Vc_it * tf.square(Vu_it) 198 | - Vf_it * Vu_it 199 | ) 200 | Vloss_bd = R["area_bd"] * tf.reduce_mean(tf.square(Vu_bd - Vu_true_bd)) 201 | 202 | Vloss = Vloss_it + R["penalty"] * Vloss_bd 203 | 204 | learning_rate = tf.placeholder_with_default(input=1e-3, shape=[], name="lr") 205 | optimizer = tf.train.AdamOptimizer(learning_rate) 206 | train_op = optimizer.minimize(Vloss) 207 | 208 | # %% train model 209 | t0 = time.time() 210 | loss, loss_bd = [], [] 211 | error = [] 212 | lr = 1 * R["learning_rate"] 213 | 214 | config = tf.ConfigProto() 215 | config.gpu_options.allow_growth = True 216 | with tf.Session(config=config) as sess: 217 | 218 | sess.run(tf.global_variables_initializer()) 219 | 220 | for epoch in range(R["total_step"] + 1): 221 | 222 | # generate new data ("N" for number) 223 | if epoch % R["resamp_epoch"] == 0: 224 | 225 | Nx_it = rand_it(R["size_it"]) 226 | Nx_bd = rand_bd(R["size_bd"]) 227 | 228 | Na_it = a(Nx_it) 229 | Nc_it = c(Nx_it) 230 | Nf_it = f(Nx_it) 231 | 232 | Nu_bd = u(Nx_bd) 233 | 234 | # train neural network 235 | lr = (1 - R["lr_decay"]) * lr 236 | 237 | _, Nloss, Nloss_bd = sess.run( 238 | [train_op, Vloss, Vloss_bd], 239 | feed_dict={ 240 | Vx_it: Nx_it, 241 | Vx_bd: Nx_bd, 242 | Va_it: Na_it, 243 | Vc_it: Nc_it, 244 | Vf_it: Nf_it, 245 | Vu_true_bd: Nu_bd, 246 | learning_rate: lr, 247 | }, 248 | ) 249 | 250 | # get test error 251 | R["u_test"] = sess.run(Vu_it, feed_dict={Vx_it: R["x_test"]}) 252 | Nerror = R["area_it"] * np.mean((R["u_test"] - R["u_test_true"]) ** 2) 253 | 254 | loss.append(Nloss) 255 | loss_bd.append(Nloss_bd) 256 | error.append(Nerror) 257 | 258 | # show current state 259 | if epoch % R["plot_epoch"] == 0: 260 | 261 | print("epoch %d, time %.3f" % (epoch, time.time() - t0)) 262 | print("total loss %f, boundary loss %f" % (Nloss, Nloss_bd)) 263 | print("interior error %f" % (Nerror)) 264 | 265 | R["time"] = time.time() - t0 266 | R["loss"] = np.array(loss) 267 | R["loss_bd"] = np.array(loss_bd) 268 | R["error"] = np.array(error) 269 | 270 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 271 | R["u_samp_" + str(epoch)] = u_samp 272 | 273 | plt.figure() 274 | ax = plt.gca() 275 | plt.semilogy(R["error"]) 276 | plt.title('error',fontsize=15) 277 | fntmp = os.path.join(R["record_path"], 'error') 278 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 279 | 280 | plt.figure() 281 | ax = plt.gca() 282 | plt.plot(R["loss"]) 283 | plt.title('loss',fontsize=15) 284 | fntmp = os.path.join(R["record_path"], 'loss') 285 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 286 | 287 | plt.figure() 288 | ax = plt.gca() 289 | plt.semilogy(R["loss_bd"]) 290 | plt.title('loss_bd',fontsize=15) 291 | fntmp = os.path.join(R["record_path"], 'loss_bd') 292 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 293 | 294 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 295 | R["u_samp_" + str(epoch)] = u_samp 296 | 297 | if R['dimension']==1: 298 | plt.figure() 299 | ax = plt.gca() 300 | plt.plot(R["x_samp"], R["u_samp_"+str(epoch)], 'r--',label='dnn') 301 | plt.plot(R["x_samp"], R["u_samp_true"], 'b-',label='true') 302 | plt.title('u',fontsize=15) 303 | plt.legend(fontsize=18) 304 | fntmp = os.path.join(R["record_path"], 'u_epoch_%s'%(epoch)) 305 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 306 | 307 | 308 | # %% save data 309 | data_dir = os.path.join(R["record_path"], "data.pkl") 310 | with open(data_dir, "wb") as file: 311 | pickle.dump(R, file) 312 | 313 | # %% save data 314 | data_dir = os.path.join(R["record_path"], "data.pkl") 315 | with open(data_dir, "wb") as file: 316 | pickle.dump(R, file) 317 | -------------------------------------------------------------------------------- /code/ritzM.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import os 8 | import time 9 | import shutil 10 | import pickle 11 | 12 | import tensorflow as tf 13 | import numpy as np 14 | import matplotlib.pyplot as plt 15 | 16 | from my_act import act_dict 17 | 18 | import matplotlib 19 | matplotlib.use('Agg') 20 | Leftp = 0.18 21 | Bottomp = 0.18 22 | Widthp = 0.88 - Leftp 23 | Heightp = 0.9 - Bottomp 24 | pos = [Leftp, Bottomp, Widthp, Heightp] 25 | 26 | def save_fig(pltm, fntmp,fp=0,ax=0,isax=0,iseps=0,isShowPic=0):# Save the figure 27 | if isax==1: 28 | pltm.rc('xtick',labelsize=18) 29 | pltm.rc('ytick',labelsize=18) 30 | ax.set_position(pos, which='both') 31 | fnm = '%s.png'%(fntmp) 32 | pltm.savefig(fnm) 33 | if iseps: 34 | fnm = '%s.eps'%(fntmp) 35 | pltm.savefig(fnm, format='eps', dpi=600) 36 | if fp!=0: 37 | fp.savefig("%s.pdf"%(fntmp), bbox_inches='tight') 38 | if isShowPic==1: 39 | pltm.show() 40 | elif isShowPic==-1: 41 | return 42 | else: 43 | pltm.close() 44 | 45 | 46 | 47 | 48 | # %% parameters 49 | R = {} 50 | 51 | R["seed"] = 0 52 | 53 | R["size_it"] = 1000 54 | R["size_bd"] = 100 55 | 56 | R["penalty"] = 1e3 57 | 58 | R["scale"] = np.array([32, 32, 32, 32, 32, 32]) 59 | 60 | R["act_name"] = ("sin",) * 3 61 | R["hidden_units"] = (30,) * 3 62 | R["learning_rate"] = 1e-4 63 | R["lr_decay"] = 5e-7 64 | 65 | R["varcoe"] = 0.5 66 | 67 | R["total_step"] = 5000 68 | R["resamp_epoch"] = 1 69 | R["plot_epoch"] = 500 70 | 71 | R["record_path"] = os.path.join("..", "exp", "sin_M",str(np.random.randint(10000,99999))) 72 | 73 | # %% generate data in domain 74 | 75 | R["dimension"] = dim = 1 76 | R["area_it"] = 2 77 | R["area_bd"] = 1 78 | 79 | # interior data 80 | def rand_it(size): 81 | x_it = np.random.rand(size, dim) * 2 - 1 82 | return x_it.astype(np.float32) 83 | 84 | 85 | # boundary data 86 | def rand_bd(size): 87 | 88 | x_bd = np.random.rand(size, dim) * 2 - 1 89 | ind = np.random.randint(dim, size=size) 90 | x01 = np.random.randint(2, size=size) * 2 - 1 91 | for ii in range(size): 92 | x_bd[ii, ind[ii]] = x01[ii] 93 | 94 | return x_bd.astype(np.float32) 95 | 96 | 97 | # %% PDE problem 98 | # - (a(x) u'(x))' + c(x) u(x) = f(x) 99 | R["mu"] = mu = 6 * np.pi 100 | 101 | 102 | def u(x): 103 | u = np.sin(mu * x) 104 | return u.astype(np.float32).reshape((-1, 1)) 105 | 106 | 107 | def a(x): 108 | a = np.ones((x.shape[0], 1)) 109 | return a.astype(np.float32).reshape((-1, 1)) 110 | 111 | 112 | def c(x): 113 | c = np.zeros((x.shape[0], 1)) 114 | return c.astype(np.float32).reshape((-1, 1)) 115 | 116 | 117 | def f(x): 118 | f = mu * mu * np.sin(mu * x) 119 | return f.astype(np.float32).reshape((-1, 1)) 120 | 121 | 122 | # %% save parameters 123 | 124 | # prepare folder 125 | if not os.path.isdir(R["record_path"]): 126 | os.mkdir(R["record_path"]) 127 | 128 | # save current code 129 | save_file = os.path.join(R["record_path"], "code.py") 130 | shutil.copyfile(__file__, save_file) 131 | 132 | # %% set seed 133 | tf.set_random_seed(R["seed"]) 134 | tf.random.set_random_seed(R["seed"]) 135 | np.random.seed(R["seed"]) 136 | 137 | # %% get test points 138 | x_test = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 139 | 140 | R["x_test"] = x_test.astype(np.float32) 141 | R["u_test_true"] = u(R["x_test"]) 142 | 143 | # %% get sample points 144 | x_samp = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 145 | 146 | R["x_samp"] = x_samp.astype(np.float32) 147 | R["u_samp_true"] = u(R["x_samp"]) 148 | 149 | # %% normal neural network 150 | units = (dim,) + R["hidden_units"] + (1,) 151 | 152 | 153 | def neural_net(x): 154 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 155 | 156 | all_y = [] 157 | for k in range(len(R["scale"])): 158 | scale_y = R["scale"][k] * x 159 | 160 | for i in range(len(units) - 2): 161 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 162 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 163 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 164 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 165 | 166 | W = tf.get_variable(name="W" + str(i) + str(k), initializer=init_W) 167 | b = tf.get_variable(name="b" + str(i) + str(k), initializer=init_b) 168 | 169 | scale_y = act_dict[R["act_name"][i]](tf.matmul(scale_y, W) + b) 170 | 171 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 172 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 173 | init_b = np.random.randn(units[-1]).astype(np.float32) 174 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 175 | 176 | W = tf.get_variable( 177 | name="W" + str(len(units) - 1) + str(k), initializer=init_W 178 | ) 179 | b = tf.get_variable( 180 | name="b" + str(len(units) - 1) + str(k), initializer=init_b 181 | ) 182 | 183 | scale_y = tf.matmul(scale_y, W) + b 184 | 185 | all_y.append(scale_y) 186 | 187 | y = sum(all_y) 188 | 189 | return y 190 | 191 | 192 | # %% loss and optimizer ("V" for variable) 193 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 194 | 195 | Vx_it = tf.placeholder(tf.float32, shape=(None, dim)) 196 | Vx_bd = tf.placeholder(tf.float32, shape=(None, dim)) 197 | 198 | Va_it = tf.placeholder(tf.float32, shape=(None, 1)) 199 | Vc_it = tf.placeholder(tf.float32, shape=(None, 1)) 200 | Vf_it = tf.placeholder(tf.float32, shape=(None, 1)) 201 | 202 | Vu_true_bd = tf.placeholder(tf.float32, shape=(None, 1)) 203 | 204 | Vu_it = neural_net(Vx_it) 205 | Vu_bd = neural_net(Vx_bd) 206 | 207 | Vdu_it = tf.gradients(Vu_it, Vx_it)[0] 208 | 209 | Vloss_it = R["area_it"] * tf.reduce_mean( 210 | 1 / 2 * Va_it * tf.reduce_sum(tf.square(Vdu_it), axis=1, keepdims=True) 211 | + 1 / 2 * Vc_it * tf.square(Vu_it) 212 | - Vf_it * Vu_it 213 | ) 214 | Vloss_bd = R["area_bd"] * tf.reduce_mean(tf.square(Vu_bd - Vu_true_bd)) 215 | 216 | Vloss = Vloss_it + R["penalty"] * Vloss_bd 217 | 218 | learning_rate = tf.placeholder_with_default(input=1e-3, shape=[], name="lr") 219 | optimizer = tf.train.AdamOptimizer(learning_rate) 220 | train_op = optimizer.minimize(Vloss) 221 | 222 | # %% train model 223 | t0 = time.time() 224 | loss, loss_bd = [], [] 225 | error = [] 226 | lr = 1 * R["learning_rate"] 227 | 228 | config = tf.ConfigProto() 229 | config.gpu_options.allow_growth = True 230 | with tf.Session(config=config) as sess: 231 | 232 | sess.run(tf.global_variables_initializer()) 233 | 234 | for epoch in range(R["total_step"] + 1): 235 | 236 | # generate new data ("N" for number) 237 | if epoch % R["resamp_epoch"] == 0: 238 | 239 | Nx_it = rand_it(R["size_it"]) 240 | Nx_bd = rand_bd(R["size_bd"]) 241 | 242 | Na_it = a(Nx_it) 243 | Nc_it = c(Nx_it) 244 | Nf_it = f(Nx_it) 245 | 246 | Nu_bd = u(Nx_bd) 247 | 248 | # train neural network 249 | lr = (1 - R["lr_decay"]) * lr 250 | 251 | _, Nloss, Nloss_bd = sess.run( 252 | [train_op, Vloss, Vloss_bd], 253 | feed_dict={ 254 | Vx_it: Nx_it, 255 | Vx_bd: Nx_bd, 256 | Va_it: Na_it, 257 | Vc_it: Nc_it, 258 | Vf_it: Nf_it, 259 | Vu_true_bd: Nu_bd, 260 | learning_rate: lr, 261 | }, 262 | ) 263 | 264 | # get test error 265 | R["u_test"] = sess.run(Vu_it, feed_dict={Vx_it: R["x_test"]}) 266 | Nerror = R["area_it"] * np.mean((R["u_test"] - R["u_test_true"]) ** 2) 267 | 268 | loss.append(Nloss) 269 | loss_bd.append(Nloss_bd) 270 | error.append(Nerror) 271 | 272 | # show current state 273 | if epoch % R["plot_epoch"] == 0: 274 | 275 | print("epoch %d, time %.3f" % (epoch, time.time() - t0)) 276 | print("total loss %f, boundary loss %f" % (Nloss, Nloss_bd)) 277 | print("interior error %f" % (Nerror)) 278 | 279 | R["time"] = time.time() - t0 280 | R["loss"] = np.array(loss) 281 | R["loss_bd"] = np.array(loss_bd) 282 | R["error"] = np.array(error) 283 | 284 | plt.figure() 285 | ax = plt.gca() 286 | plt.semilogy(R["error"]) 287 | plt.title('error',fontsize=15) 288 | fntmp = os.path.join(R["record_path"], 'error') 289 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 290 | 291 | plt.figure() 292 | ax = plt.gca() 293 | plt.plot(R["loss"]) 294 | plt.title('loss',fontsize=15) 295 | fntmp = os.path.join(R["record_path"], 'loss') 296 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 297 | 298 | plt.figure() 299 | ax = plt.gca() 300 | plt.semilogy(R["loss_bd"]) 301 | plt.title('loss_bd',fontsize=15) 302 | fntmp = os.path.join(R["record_path"], 'loss_bd') 303 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 304 | 305 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 306 | R["u_samp_" + str(epoch)] = u_samp 307 | 308 | if R['dimension']==1: 309 | plt.figure() 310 | ax = plt.gca() 311 | plt.plot(R["x_samp"], R["u_samp_"+str(epoch)], 'r--',label='dnn') 312 | plt.plot(R["x_samp"], R["u_samp_true"], 'b-',label='true') 313 | plt.title('u',fontsize=15) 314 | plt.legend(fontsize=18) 315 | fntmp = os.path.join(R["record_path"], 'u_epoch_%s'%(epoch)) 316 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 317 | 318 | 319 | 320 | 321 | # %% save data 322 | data_dir = os.path.join(R["record_path"], "data.pkl") 323 | with open(data_dir, "wb") as file: 324 | pickle.dump(R, file) 325 | 326 | # %% save data 327 | data_dir = os.path.join(R["record_path"], "data.pkl") 328 | with open(data_dir, "wb") as file: 329 | pickle.dump(R, file) 330 | -------------------------------------------------------------------------------- /exp/sin_M/62178/code.py: -------------------------------------------------------------------------------- 1 | # !python3 2 | # coding: utf-8 3 | # author: Ziqi Liu and Zhi-Qin John Xu 4 | # Reference: Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) 5 | # for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 2020. arXiv:2007.11207 (CiCP) 6 | 7 | import os 8 | import time 9 | import shutil 10 | import pickle 11 | 12 | import tensorflow as tf 13 | import numpy as np 14 | import matplotlib.pyplot as plt 15 | 16 | from my_act import act_dict 17 | 18 | import matplotlib 19 | matplotlib.use('Agg') 20 | Leftp = 0.18 21 | Bottomp = 0.18 22 | Widthp = 0.88 - Leftp 23 | Heightp = 0.9 - Bottomp 24 | pos = [Leftp, Bottomp, Widthp, Heightp] 25 | 26 | def save_fig(pltm, fntmp,fp=0,ax=0,isax=0,iseps=0,isShowPic=0):# Save the figure 27 | if isax==1: 28 | pltm.rc('xtick',labelsize=18) 29 | pltm.rc('ytick',labelsize=18) 30 | ax.set_position(pos, which='both') 31 | fnm = '%s.png'%(fntmp) 32 | pltm.savefig(fnm) 33 | if iseps: 34 | fnm = '%s.eps'%(fntmp) 35 | pltm.savefig(fnm, format='eps', dpi=600) 36 | if fp!=0: 37 | fp.savefig("%s.pdf"%(fntmp), bbox_inches='tight') 38 | if isShowPic==1: 39 | pltm.show() 40 | elif isShowPic==-1: 41 | return 42 | else: 43 | pltm.close() 44 | 45 | 46 | 47 | 48 | # %% parameters 49 | R = {} 50 | 51 | R["seed"] = 0 52 | 53 | R["size_it"] = 1000 54 | R["size_bd"] = 100 55 | 56 | R["penalty"] = 1e3 57 | 58 | R["scale"] = np.array([32, 32, 32, 32, 32, 32]) 59 | 60 | R["act_name"] = ("sin",) * 3 61 | R["hidden_units"] = (30,) * 3 62 | R["learning_rate"] = 1e-4 63 | R["lr_decay"] = 5e-7 64 | 65 | R["varcoe"] = 0.5 66 | 67 | R["total_step"] = 5000 68 | R["resamp_epoch"] = 1 69 | R["plot_epoch"] = 500 70 | 71 | R["record_path"] = os.path.join("..", "exp", "sin_M",str(np.random.randint(10000,99999))) 72 | 73 | # %% generate data in domain 74 | 75 | R["dimension"] = dim = 1 76 | R["area_it"] = 2 77 | R["area_bd"] = 1 78 | 79 | # interior data 80 | def rand_it(size): 81 | x_it = np.random.rand(size, dim) * 2 - 1 82 | return x_it.astype(np.float32) 83 | 84 | 85 | # boundary data 86 | def rand_bd(size): 87 | 88 | x_bd = np.random.rand(size, dim) * 2 - 1 89 | ind = np.random.randint(dim, size=size) 90 | x01 = np.random.randint(2, size=size) * 2 - 1 91 | for ii in range(size): 92 | x_bd[ii, ind[ii]] = x01[ii] 93 | 94 | return x_bd.astype(np.float32) 95 | 96 | 97 | # %% PDE problem 98 | # - (a(x) u'(x))' + c(x) u(x) = f(x) 99 | R["mu"] = mu = 6 * np.pi 100 | 101 | 102 | def u(x): 103 | u = np.sin(mu * x) 104 | return u.astype(np.float32).reshape((-1, 1)) 105 | 106 | 107 | def a(x): 108 | a = np.ones((x.shape[0], 1)) 109 | return a.astype(np.float32).reshape((-1, 1)) 110 | 111 | 112 | def c(x): 113 | c = np.zeros((x.shape[0], 1)) 114 | return c.astype(np.float32).reshape((-1, 1)) 115 | 116 | 117 | def f(x): 118 | f = mu * mu * np.sin(mu * x) 119 | return f.astype(np.float32).reshape((-1, 1)) 120 | 121 | 122 | # %% save parameters 123 | 124 | # prepare folder 125 | if not os.path.isdir(R["record_path"]): 126 | os.mkdir(R["record_path"]) 127 | 128 | # save current code 129 | save_file = os.path.join(R["record_path"], "code.py") 130 | shutil.copyfile(__file__, save_file) 131 | 132 | # %% set seed 133 | tf.set_random_seed(R["seed"]) 134 | tf.random.set_random_seed(R["seed"]) 135 | np.random.seed(R["seed"]) 136 | 137 | # %% get test points 138 | x_test = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 139 | 140 | R["x_test"] = x_test.astype(np.float32) 141 | R["u_test_true"] = u(R["x_test"]) 142 | 143 | # %% get sample points 144 | x_samp = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 145 | 146 | R["x_samp"] = x_samp.astype(np.float32) 147 | R["u_samp_true"] = u(R["x_samp"]) 148 | 149 | # %% normal neural network 150 | units = (dim,) + R["hidden_units"] + (1,) 151 | 152 | 153 | def neural_net(x): 154 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 155 | 156 | all_y = [] 157 | for k in range(len(R["scale"])): 158 | scale_y = R["scale"][k] * x 159 | 160 | for i in range(len(units) - 2): 161 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 162 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 163 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 164 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 165 | 166 | W = tf.get_variable(name="W" + str(i) + str(k), initializer=init_W) 167 | b = tf.get_variable(name="b" + str(i) + str(k), initializer=init_b) 168 | 169 | scale_y = act_dict[R["act_name"][i]](tf.matmul(scale_y, W) + b) 170 | 171 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 172 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 173 | init_b = np.random.randn(units[-1]).astype(np.float32) 174 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 175 | 176 | W = tf.get_variable( 177 | name="W" + str(len(units) - 1) + str(k), initializer=init_W 178 | ) 179 | b = tf.get_variable( 180 | name="b" + str(len(units) - 1) + str(k), initializer=init_b 181 | ) 182 | 183 | scale_y = tf.matmul(scale_y, W) + b 184 | 185 | all_y.append(scale_y) 186 | 187 | y = sum(all_y) 188 | 189 | return y 190 | 191 | 192 | # %% loss and optimizer ("V" for variable) 193 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 194 | 195 | Vx_it = tf.placeholder(tf.float32, shape=(None, dim)) 196 | Vx_bd = tf.placeholder(tf.float32, shape=(None, dim)) 197 | 198 | Va_it = tf.placeholder(tf.float32, shape=(None, 1)) 199 | Vc_it = tf.placeholder(tf.float32, shape=(None, 1)) 200 | Vf_it = tf.placeholder(tf.float32, shape=(None, 1)) 201 | 202 | Vu_true_bd = tf.placeholder(tf.float32, shape=(None, 1)) 203 | 204 | Vu_it = neural_net(Vx_it) 205 | Vu_bd = neural_net(Vx_bd) 206 | 207 | Vdu_it = tf.gradients(Vu_it, Vx_it)[0] 208 | 209 | Vloss_it = R["area_it"] * tf.reduce_mean( 210 | 1 / 2 * Va_it * tf.reduce_sum(tf.square(Vdu_it), axis=1, keepdims=True) 211 | + 1 / 2 * Vc_it * tf.square(Vu_it) 212 | - Vf_it * Vu_it 213 | ) 214 | Vloss_bd = R["area_bd"] * tf.reduce_mean(tf.square(Vu_bd - Vu_true_bd)) 215 | 216 | Vloss = Vloss_it + R["penalty"] * Vloss_bd 217 | 218 | learning_rate = tf.placeholder_with_default(input=1e-3, shape=[], name="lr") 219 | optimizer = tf.train.AdamOptimizer(learning_rate) 220 | train_op = optimizer.minimize(Vloss) 221 | 222 | # %% train model 223 | t0 = time.time() 224 | loss, loss_bd = [], [] 225 | error = [] 226 | lr = 1 * R["learning_rate"] 227 | 228 | config = tf.ConfigProto() 229 | config.gpu_options.allow_growth = True 230 | with tf.Session(config=config) as sess: 231 | 232 | sess.run(tf.global_variables_initializer()) 233 | 234 | for epoch in range(R["total_step"] + 1): 235 | 236 | # generate new data ("N" for number) 237 | if epoch % R["resamp_epoch"] == 0: 238 | 239 | Nx_it = rand_it(R["size_it"]) 240 | Nx_bd = rand_bd(R["size_bd"]) 241 | 242 | Na_it = a(Nx_it) 243 | Nc_it = c(Nx_it) 244 | Nf_it = f(Nx_it) 245 | 246 | Nu_bd = u(Nx_bd) 247 | 248 | # train neural network 249 | lr = (1 - R["lr_decay"]) * lr 250 | 251 | _, Nloss, Nloss_bd = sess.run( 252 | [train_op, Vloss, Vloss_bd], 253 | feed_dict={ 254 | Vx_it: Nx_it, 255 | Vx_bd: Nx_bd, 256 | Va_it: Na_it, 257 | Vc_it: Nc_it, 258 | Vf_it: Nf_it, 259 | Vu_true_bd: Nu_bd, 260 | learning_rate: lr, 261 | }, 262 | ) 263 | 264 | # get test error 265 | R["u_test"] = sess.run(Vu_it, feed_dict={Vx_it: R["x_test"]}) 266 | Nerror = R["area_it"] * np.mean((R["u_test"] - R["u_test_true"]) ** 2) 267 | 268 | loss.append(Nloss) 269 | loss_bd.append(Nloss_bd) 270 | error.append(Nerror) 271 | 272 | # show current state 273 | if epoch % R["plot_epoch"] == 0: 274 | 275 | print("epoch %d, time %.3f" % (epoch, time.time() - t0)) 276 | print("total loss %f, boundary loss %f" % (Nloss, Nloss_bd)) 277 | print("interior error %f" % (Nerror)) 278 | 279 | R["time"] = time.time() - t0 280 | R["loss"] = np.array(loss) 281 | R["loss_bd"] = np.array(loss_bd) 282 | R["error"] = np.array(error) 283 | 284 | plt.figure() 285 | ax = plt.gca() 286 | plt.semilogy(R["error"]) 287 | plt.title('error',fontsize=15) 288 | fntmp = os.path.join(R["record_path"], 'error') 289 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 290 | 291 | plt.figure() 292 | ax = plt.gca() 293 | plt.plot(R["loss"]) 294 | plt.title('loss',fontsize=15) 295 | fntmp = os.path.join(R["record_path"], 'loss') 296 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 297 | 298 | plt.figure() 299 | ax = plt.gca() 300 | plt.semilogy(R["loss_bd"]) 301 | plt.title('loss_bd',fontsize=15) 302 | fntmp = os.path.join(R["record_path"], 'loss_bd') 303 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 304 | 305 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 306 | R["u_samp_" + str(epoch)] = u_samp 307 | 308 | if R['dimension']==1: 309 | plt.figure() 310 | ax = plt.gca() 311 | plt.plot(R["x_samp"], R["u_samp_"+str(epoch)], 'r--',label='dnn') 312 | plt.plot(R["x_samp"], R["u_samp_true"], 'b-',label='true') 313 | plt.title('u',fontsize=15) 314 | plt.legend(fontsize=18) 315 | fntmp = os.path.join(R["record_path"], 'u_epoch_%s'%(epoch)) 316 | save_fig(plt,fntmp,ax=ax,isax=1,iseps=0) 317 | 318 | 319 | 320 | 321 | # %% save data 322 | data_dir = os.path.join(R["record_path"], "data.pkl") 323 | with open(data_dir, "wb") as file: 324 | pickle.dump(R, file) 325 | 326 | # %% save data 327 | data_dir = os.path.join(R["record_path"], "data.pkl") 328 | with open(data_dir, "wb") as file: 329 | pickle.dump(R, file) 330 | -------------------------------------------------------------------------------- /notes/notes.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt,a4paper]{article} 2 | 3 | \usepackage[utf8]{inputenc} 4 | \usepackage{ctex} 5 | \usepackage{amsmath,amsfonts,amssymb} 6 | \usepackage{abstract,appendix} 7 | \usepackage{makeidx,hyperref} 8 | \usepackage{graphicx,epsfig,subfig} 9 | \usepackage{geometry} 10 | \usepackage{color,xcolor} 11 | \usepackage{listings} 12 | 13 | \lstset{ 14 | basicstyle=\footnotesize, % 设定代码字体 15 | backgroundcolor=\color[RGB]{239,239,239}, % 设定背景颜色 16 | keywordstyle=\color[RGB]{40,40,255}, % 设定关键字颜色 17 | commentstyle=\color[RGB]{0,96,96}, % 设置代码注释的格式 18 | stringstyle=\color[RGB]{128,0,0}, % 设置字符串格式 19 | language=python, % 设置语言 20 | } 21 | 22 | \geometry{scale=0.8} 23 | 24 | %\setlength{\lineskip}{\baselineskip} 25 | \setlength{\parskip}{0.5\baselineskip} 26 | 27 | \title{深度学习（Mscale网络）求解椭圆方程代码模板（Deep-Ritz方法）} 28 | \author{刘子旗, 许志钦\thanks{xuzhiqin@sjtu.edu.cn}} 29 | \date{\today} 30 | 31 | \begin{document} 32 | 33 | \maketitle 34 | 35 | 根据神经网络的F-Principle \cite{fprinciple}，网络在训练的过程中会优先拟合训练数据的低频部分。但是在某些特殊的场合下（如使用神经网络求解微分方程），需要拟合目标函数的高频部分。实验希望找到一种特殊的网络结构，更快地拟合目标函数的高频部分。 36 | 37 | 这里提出了一种新的网络结构：Mscale，用于求解椭圆方程。 38 | 39 | \section*{椭圆方程的Ritz方法} 40 | 41 | 椭圆方程的一般形式为 42 | \begin{align*} 43 | -\nabla(a(x) \nabla u(x)) + c(x) u(x) = f(x) \quad x \in \Omega \\ 44 | u(x) = u_0(x) \quad x \in \partial \Omega 45 | \end{align*} 46 | 47 | 通过数学上的推导\cite{deep-ritz}，方程可以转化为泛函的极小值问题 48 | \begin{align*} 49 | \min_{u} J(u) & = \int_{\Omega} \frac{1}{2} a(x) |\nabla u(x)|^2 + c(x) |u(x)|^2 - f(x) u(x) \ dx \\ 50 | \text{s.t.} \quad u(x) & = u_0(x) \quad x \in \partial \Omega 51 | \end{align*} 52 | 53 | 这是一个约束优化问题，我们用加惩罚项的方法把它近似转化成无约束优化问题 54 | \begin{align*} 55 | \min_{u} \int_{\Omega} \frac{1}{2} a(x) |\nabla u(x)|^2 + \frac{1}{2} c(x) |u(x)|^2 - f(x) u(x) \ dx + \int_{\partial \Omega} |u(x) - u_0(x)|^2 \ dx 56 | \end{align*} 57 | 58 | 用神经网络$N(x; \theta)$表示未知函数，用蒙特卡洛方法计算积分的值，就得到了最终的优化问题 59 | \begin{align*} 60 | \min_{\theta} L(\theta) & = \frac{|\Omega|}{|S_1|} \sum_{x_i \in S_1} \frac{1}{2} a(x_i) |\nabla N(x_i; \theta)|^2 + \frac{1}{2} c(x_i) |N(x_i; \theta)|^2 - f(x_i) N(x_i; \theta) \\ 61 | & + \beta \frac{|\partial \Omega|}{|S_2|} \sum_{x_i \in S_2} |N(x_i; \theta) - u_0(x_i)|^2 \ dx 62 | \end{align*} 63 | 其中$S_1$和$S_2$是在$\Omega$和$\partial \Omega$上随机生成的点。 64 | 65 | \section*{神经网络结构} 66 | 67 | 一般的神经网络定义为一个函数$N(x): \mathbb{R}^d \rightarrow \mathbb{R}$，由线性变换和非线性变换复合而成。 68 | \begin{align*} 69 | N(x; \theta) = W^{[L]} \sigma(\cdots W^{[2]} \sigma(W^{[1]} x + b^{[1]}) + b^{[2]} \cdots) + b^{[L]} 70 | \end{align*} 71 | 其中$\sigma: \mathbb{R} \rightarrow \mathbb{R}$叫做“激活函数”，它作用在向量上相当于作用于每个分量。$L$叫做“层数”，$W^{[l]}, b^{[l]}$是待定的参数。 72 | 73 | 其中$n^{[l]}$叫做每一层的“宽度”，满足$n^{[1]} = d, n^{[L+1]} = 1$ 74 | \begin{align*} 75 | W^{[l]} \in \mathbb{R}^{n^{[l]} \times n^{[l+1]}}, \quad b^{[l]} \in \mathbb{R}^{n^{[l+1]}} 76 | \end{align*} 77 | 78 | \section*{Mscale网络结构} 79 | 80 | 网络结构\cite{mscale}如图\ref{net}。具体代码实现见下文。 81 | 82 | \begin{figure}[h] 83 | \centering 84 | \includegraphics[width=0.5\linewidth]{MscaleNet2} 85 | \caption{Mscale网络结构} 86 | \label{net} 87 | \end{figure} 88 | 89 | \section*{代码实现细节} 90 | 91 | 代码文件说明： 92 | 93 | \begin{itemize} 94 | \item \textbf{ritzN.py}：用一般的Normal网络求解椭圆方程。 95 | \item \textbf{ritzM.py}：用Mscale网络求解椭圆方程。 96 | \item \textbf{plot\_result.py}：展示网络结果，画图。 97 | \item \textbf{my\_act.py}：文件中定义了可能用到的激活函数，如果要用到新的激活函数可以在这里添加。\textbf{act\_dict}变量下标是激活函数的“名字”，它的值是名字对应的函数句柄。 98 | \end{itemize} 99 | 100 | \textbf{运行环境：python 3.6 + tensorflow 1.16} 101 | 102 | \subsection*{参数设定} 103 | 104 | 在代码中，首先我们要设定所有的参数，以便修改。所有的参数和运行结果都记录在变量\textbf{R}中。 105 | 106 | \begin{lstlisting} 107 | R = {} 108 | 109 | R["seed"] = 0 110 | 111 | R["size_it"] = 1000 112 | R["size_bd"] = 100 113 | 114 | R["penalty"] = 1e3 115 | 116 | R["act_name"] = ("sin",) * 3 117 | R["hidden_units"] = (180,) * 3 118 | R["learning_rate"] = 1e-4 119 | R["lr_decay"] = 5e-7 120 | 121 | R["varcoe"] = 0.5 122 | 123 | R["total_step"] = 5000 124 | R["resamp_epoch"] = 1 125 | R["plot_epoch"] = 500 126 | 127 | R["record_path"] = os.path.join("..", "exp", "sin_N") 128 | \end{lstlisting} 129 | 130 | 参数意义如下： 131 | 132 | \begin{itemize} 133 | \item \textbf{seed}：随机数种子。保证两次运行代码的结果完全一样。（在必要的时候，这个也是可调的参数之一） 134 | \item \textbf{size\_it}：区域$\Omega$内部随机生成的点的个数，也就是$|S_1|$。 135 | \item \textbf{size\_bd}：区域边界$\partial \Omega$上随机生成的点的个数，也就是$|S_2|$。 136 | \item \textbf{penalty}：惩罚系数，就是公式中的$\beta$。 137 | \item \textbf{scale}：Mscale网络中的变换系数。（Normal网络中没有这个参数） 138 | \item \textbf{act\_name}：长度为$N$的列表，代表每一层的激活函数。 139 | \item \textbf{hidden\_units}：长度为$N$的列表，代表每一层的宽度。 140 | \item \textbf{learning\_rate}：学习率。（一般不能太大） 141 | \item \textbf{lr\_decay}：学习率下降速度。 142 | \item \textbf{varcoe}：初始化参数。 143 | \item \textbf{total\_step}：训练总步数。 144 | \item \textbf{resamp\_epoch}：采样间隔。每隔X步重新随机生成$S_1,S_2$中的点。X=1代表每计算一步都重新随机生成点。X大于总步数时，代表训练用的样本不变。 145 | \item \textbf{plot\_epoch}：从第0步开始，每隔X步记录下数值解图像。同时输出当前状态。 146 | \item \textbf{record\_path}：保存记录的文件夹路径。在运行结束的时候，源代码文件会被复制到文件夹下的\textbf{code.py}文件中，以便重复实验。数据结果和参数会保存在文件夹下的\textbf{data.pkl}文件中，以便画图和展示。 147 | \end{itemize} 148 | 149 | \subsection*{定义求解区域} 150 | 151 | 这一部分定义了求解区域$\Omega$。\textbf{如果要求解复杂区域上的方程，就要修改这部分代码。} 152 | 153 | \begin{lstlisting} 154 | R["dimension"] = dim = 2 155 | R["area_it"] = 4 156 | R["area_bd"] = 8 157 | 158 | # interior data 159 | def rand_it(size): 160 | x_it = np.random.rand(size, dim) * 2 - 1 161 | return x_it.astype(np.float32) 162 | 163 | # boundary data 164 | def rand_bd(size): 165 | 166 | x_bd = np.random.rand(size, dim) * 2 - 1 167 | ind = np.random.randint(dim, size=size) 168 | x01 = np.random.randint(2, size=size) * 2 - 1 169 | for ii in range(size): 170 | x_bd[ii, ind[ii]] = x01[ii] 171 | return x_bd.astype(np.float32) 172 | \end{lstlisting} 173 | 174 | 这里\textbf{dim}是方程的空间维数，\textbf{area\_it}表示$\Omega$的面积。\textbf{area\_bd}表示$\partial \Omega$的面积。 175 | 176 | \textbf{rand\_it}函数，生成$\Omega$内\textbf{均匀分布}的随机点，输入需要的随机点个数\textbf{size}，返回一个大小为$size \times dim$的数组。 177 | 178 | \textbf{rand\_bd}函数，生成边界$\partial \Omega$上\textbf{均匀分布}的随机点，输入输出同上。 179 | 180 | 这样就定义了求解区域。这里对应的求解区域是$[-1,1]^2$。想要在其它区域上求解，只要重新实现一遍这个函数。 181 | 182 | \subsection*{定义方程} 183 | 184 | 这一部分定义了方程中的已知函数$a(x),c(x),f(x)$，和精确解（边界条件）$u(x)$。如果精确解未知，$u(x)$就代表边界条件。\textbf{如果要求解其它方程，就要修改这部分代码。} 185 | 186 | \begin{lstlisting} 187 | # PDE problem 188 | # - (a(x) u'(x))' + c(x) u(x) = f(x) 189 | R["mu"] = mu = 6 * np.pi 190 | 191 | def u(x): 192 | u = np.sin(mu * x) 193 | return u.astype(np.float32).reshape((-1, 1)) 194 | 195 | def a(x): 196 | a = np.ones((x.shape[0], 1)) 197 | return a.astype(np.float32).reshape((-1, 1)) 198 | 199 | def c(x): 200 | c = np.zeros((x.shape[0], 1)) 201 | return c.astype(np.float32).reshape((-1, 1)) 202 | 203 | def f(x): 204 | f = mu * mu * np.sin(mu * x) 205 | return f.astype(np.float32).reshape((-1, 1)) 206 | \end{lstlisting} 207 | 208 | 需要注意的是：这些函数，输入都是一个大小为$size \times dim$的数组，每一行代表一个采样点。返回的是大小为$size \times 1$的数组。（这里的维度不一致会报错） 209 | 210 | \subsection*{保存参数} 211 | 212 | 把源代码文件复制到文件夹下的\textbf{code.py}文件中，以便重复实验。设定随机数种子。 213 | 214 | \begin{lstlisting} 215 | # save parameters 216 | 217 | # prepare folder 218 | if not os.path.isdir(R["record_path"]): 219 | os.mkdir(R["record_path"]) 220 | 221 | # save current code 222 | save_file = os.path.join(R["record_path"], "code.py") 223 | shutil.copyfile(__file__, save_file) 224 | 225 | # set seed 226 | tf.set_random_seed(R["seed"]) 227 | tf.random.set_random_seed(R["seed"]) 228 | np.random.seed(R["seed"]) 229 | \end{lstlisting} 230 | 231 | \subsection*{用于画图和计算误差的采样点} 232 | 233 | 这里的\textbf{x\_test}是用于计算误差的采样点，\textbf{u\_test\_true}是这些点上的精确解的值。计算数值解和精确解误差的时候就在这些点上计算。 234 | 235 | \textbf{x\_samp}是用于画图的采样点。\textbf{u\_samp\_true}是这些点上的精确解的值。 236 | 237 | \textbf{如果我们不知道精确解，而是通过差分方法或者其它方法生成的参考解。这里的\textbf{u\_test\_true}就要从文件里面读取出来。} 238 | 239 | \begin{lstlisting} 240 | # get test points 241 | x_test = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 242 | 243 | R["x_test"] = x_test.astype(np.float32) 244 | R["u_test_true"] = u(R["x_test"]) 245 | 246 | # get sample points 247 | x_samp = np.linspace(-1, 1, 200 +1).reshape((-1, 1)) 248 | 249 | R["x_samp"] = x_samp.astype(np.float32) 250 | R["u_samp_true"] = u(R["x_samp"]) 251 | \end{lstlisting} 252 | 253 | 需要注意的是：这里的\textbf{x\_test}和\textbf{x\_samp}都是$size \times dim$的数组，每一行代表一个采样点。这里的\textbf{u\_test\_true}和\textbf{u\_samp\_true}都是$size \times 1$的数组。在画图时要另外调整。 254 | 255 | 之后在训练的过程中，还会有\textbf{u\_test\_X}和\textbf{u\_samp\_X}变量，代表训练到第X步的数值解。 256 | 257 | \subsection*{神经网络} 258 | 259 | 用tensorflow实现深度学习的网络结构。 260 | 261 | \textbf{如果要换网络结构或者参数初始化，就要改这段代码。} 262 | 263 | 下面这段代码实现了一个普通的网络（Normal），\textbf{init\_W}和\textbf{init\_b}是变量的初始值。 264 | 265 | \begin{lstlisting} 266 | units = (dim,) + R["hidden_units"] + (1,) 267 | 268 | def neural_net(x): 269 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 270 | 271 | y = x 272 | 273 | for i in range(len(units) - 2): 274 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 275 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 276 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 277 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 278 | 279 | W = tf.get_variable(name="W" + str(i), initializer=init_W) 280 | b = tf.get_variable(name="b" + str(i), initializer=init_b) 281 | 282 | y = act_dict[R["act_name"][i]](tf.matmul(y, W) + b) 283 | 284 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 285 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 286 | init_b = np.random.randn(units[-1]).astype(np.float32) 287 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 288 | 289 | W = tf.get_variable(name="W" + str(len(units) - 1), initializer=init_W) 290 | b = tf.get_variable(name="b" + str(len(units) - 1), initializer=init_b) 291 | 292 | y = tf.matmul(y, W) + b 293 | 294 | return y 295 | \end{lstlisting} 296 | 297 | 下面这段代码实现了Mscale网络，原理类似。 298 | 299 | \begin{lstlisting} 300 | units = (dim,) + R["hidden_units"] + (1,) 301 | 302 | def neural_net(x): 303 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 304 | 305 | all_y = [] 306 | for k in range(len(R["scale"])): 307 | scale_y = R["scale"][k] * x 308 | 309 | for i in range(len(units) - 2): 310 | init_W = np.random.randn(units[i], units[i + 1]).astype(np.float32) 311 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 312 | init_b = np.random.randn(units[i + 1]).astype(np.float32) 313 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 314 | 315 | W = tf.get_variable(name="W" + str(i) + str(k), 316 | initializer=init_W) 317 | b = tf.get_variable(name="b" + str(i) + str(k), 318 | initializer=init_b) 319 | 320 | scale_y = act_dict[R["act_name"][i]](tf.matmul(scale_y, W) + b) 321 | 322 | init_W = np.random.randn(units[-2], units[-1]).astype(np.float32) 323 | init_W = init_W * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 324 | init_b = np.random.randn(units[-1]).astype(np.float32) 325 | init_b = init_b * (2 / (units[i] + units[i + 1]) ** R["varcoe"]) 326 | 327 | W = tf.get_variable(name="W" + str(len(units) - 1) + str(k), 328 | initializer=init_W) 329 | b = tf.get_variable(name="b" + str(len(units) - 1) + str(k), 330 | initializer=init_b) 331 | 332 | scale_y = tf.matmul(scale_y, W) + b 333 | 334 | all_y.append(scale_y) 335 | 336 | y = sum(all_y) 337 | 338 | return y 339 | \end{lstlisting} 340 | 341 | \subsection*{损失函数} 342 | 343 | 定义损失函数。这里变量前面的“V”代表它的类型是tensorflow中的Variable，变量前面的“N”表示它保存变量当前的取值。 344 | 345 | \textbf{如果求解的不是椭圆方程，或者损失函数的形式有变化，就要改这段代码。} 346 | 347 | \begin{lstlisting} 348 | # loss and optimizer ("V" for variable) 349 | with tf.variable_scope("vscope", reuse=tf.AUTO_REUSE): 350 | 351 | Vx_it = tf.placeholder(tf.float32, shape=(None, dim)) 352 | Vx_bd = tf.placeholder(tf.float32, shape=(None, dim)) 353 | 354 | Va_it = tf.placeholder(tf.float32, shape=(None, 1)) 355 | Vc_it = tf.placeholder(tf.float32, shape=(None, 1)) 356 | Vf_it = tf.placeholder(tf.float32, shape=(None, 1)) 357 | 358 | Vu_true_bd = tf.placeholder(tf.float32, shape=(None, 1)) 359 | 360 | Vu_it = neural_net(Vx_it) 361 | Vu_bd = neural_net(Vx_bd) 362 | 363 | Vdu_it = tf.gradients(Vu_it, Vx_it)[0] 364 | 365 | Vloss_it = R["area_it"] * tf.reduce_mean( 366 | 1 / 2 * Va_it * tf.reduce_sum(tf.square(Vdu_it), axis=1, keepdims=True) 367 | + 1 / 2 * Vc_it * tf.square(Vu_it) 368 | - Vf_it * Vu_it 369 | ) 370 | Vloss_bd = R["area_bd"] * tf.reduce_mean(tf.square(Vu_bd - Vu_true_bd)) 371 | 372 | Vloss = Vloss_it + R["penalty"] * Vloss_bd 373 | 374 | learning_rate = tf.placeholder_with_default(input=1e-3, shape=[], name="lr") 375 | optimizer = tf.train.AdamOptimizer(learning_rate) 376 | train_op = optimizer.minimize(Vloss) 377 | \end{lstlisting} 378 | 379 | \subsection*{开始训练} 380 | 381 | \textbf{这里的代码一般不用大改，但是如果前面有改动，这里的要和之前的匹配。} 382 | 383 | 随机生成定义域里的点，计算这些点上的函数值。 384 | \begin{lstlisting} 385 | # generate new data ("N" for number) 386 | if epoch % R["resamp_epoch"] == 0: 387 | Nx_it = rand_it(R["size_it"]) 388 | Nx_bd = rand_bd(R["size_bd"]) 389 | Na_it = a(Nx_it) 390 | Nc_it = c(Nx_it) 391 | Nf_it = f(Nx_it) 392 | Nu_bd = u(Nx_bd) 393 | \end{lstlisting} 394 | 395 | 把生成的数据喂给网络，训练一步。 396 | \begin{lstlisting} 397 | _, Nloss, Nloss_bd = sess.run( 398 | [train_op, Vloss, Vloss_bd], 399 | feed_dict={ 400 | Vx_it: Nx_it, 401 | Vx_bd: Nx_bd, 402 | Va_it: Na_it, 403 | Vc_it: Nc_it, 404 | Vf_it: Nf_it, 405 | Vu_true_bd: Nu_bd, 406 | learning_rate: lr, 407 | }, 408 | ) 409 | \end{lstlisting} 410 | 411 | 代入采样点，计算数值解和真实解之间的误差 412 | \begin{lstlisting} 413 | R["u_test"] = sess.run(Vu_it, feed_dict={Vx_it: R["x_test"]}) 414 | Nerror = R["area_it"] * np.mean((R["u_test"] - R["u_test_true"]) ** 2) 415 | \end{lstlisting} 416 | 417 | 每过\textbf{plot\_epoch}步，记录下当时的数值解，同时输出当前误差。 418 | \begin{lstlisting} 419 | if epoch % R["plot_epoch"] == 0: 420 | print("epoch %d, time %.3f" % (epoch, time.time() - t0)) 421 | print("total loss %f, boundary loss %f" % (Nloss, Nloss_bd)) 422 | print("interior error %f" % (Nerror)) 423 | 424 | u_samp = sess.run(Vu_it, feed_dict={Vx_it: R["x_samp"]}) 425 | R["u_samp_" + str(epoch)] = u_samp 426 | \end{lstlisting} 427 | 428 | 最后保存所有参数 429 | \begin{lstlisting} 430 | # save data 431 | data_dir = os.path.join(R["record_path"], "data.pkl") 432 | with open(data_dir, "wb") as file: 433 | pickle.dump(R, file) 434 | \end{lstlisting} 435 | 436 | \section*{测试用例} 437 | 438 | 具体参数如上，分别用规模为(180,180,180)的普通全连接网络(Normal)和系数为(1,2,4,8,16,32)的6个规模为(30,30,30)的网络(Mscale)求解方程。激活函数为sin(x). 439 | 440 | 方程的精确解为$u = \sin 6 \pi x$。 441 | 442 | (mscale1-32)求解过程中，数值解随训练的变化如图\ref{proc}。蓝色实线是精确解，红色虚线是数值解。 443 | \begin{figure}[h] 444 | \centering 445 | \subfloat[0步]{\includegraphics[width=0.24\linewidth]{s0}} 446 | \subfloat[500步]{\includegraphics[width=0.24\linewidth]{s500}} 447 | \subfloat[1000步]{\includegraphics[width=0.24\linewidth]{s1000}} 448 | \subfloat[5000步]{\includegraphics[width=0.24\linewidth]{s5000}} 449 | \caption{求解过程} 450 | \label{proc} 451 | \end{figure} 452 | 453 | 两种方法error下降对比如图\ref{error}。横轴是训练步数，纵轴是测试误差。 454 | \begin{figure}[h] 455 | \centering 456 | \includegraphics[width=0.3\linewidth]{error} 457 | \caption{误差下降曲线} 458 | \label{error} 459 | \end{figure} 460 | 461 | \begin{thebibliography}{99} 462 | \bibitem{fprinciple} Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks. Communications in 463 | Computational Physics, 2020, arXiv: 1901.06523. 464 | \bibitem{deep-ritz} Weinan E ,Yu B. The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems[J]. Communications in Mathematics \& Stats, 2018, 6(1):1-12. 465 | \bibitem{mscale}Ziqi Liu，Wei Cai，Zhi-Qin John Xu. Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains[J]. 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