1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
|
import torch
import torch.optim as optim
import numpy as np
import matplotlib.pyplot as plt
class VanillaRNN(torch.nn.Module):
def __init__(self, Nin, N, Nout):
super(VanillaRNN, self).__init__()
self.N = N
self.W_in = torch.nn.Parameter(torch.randn(N, Nin) / np.sqrt(Nin))
self.W_h = torch.nn.Parameter(torch.randn(N, N) / np.sqrt(N))
self.b_h = torch.nn.Parameter(torch.zeros(N))
self.W_out = torch.nn.Parameter(torch.randn(Nout, N) / np.sqrt(N))
self.b_out = torch.nn.Parameter(torch.zeros(Nout))
def forward(self, x, h_prev):
h = torch.tanh(self.W_in @ x + self.W_h @ h_prev + self.b_h)
y = self.W_out @ h + self.b_out
return h, y
def calculate_jacobian_analytical(vanilla_rnn, h):
tanh_prime = 1 / torch.cosh(h)**2
jacobian = vanilla_rnn.W_h @ torch.diag(tanh_prime)
return jacobian
def calculate_lyapunov_spectrum(vanilla_rnn, x_data, nle, seedIC=1, seedONS=1):
n = vanilla_rnn.N
steps = len(x_data)
ONSstep = 1
torch.manual_seed(seedIC)
h = torch.zeros(n, dtype=torch.float32, requires_grad=True)
torch.manual_seed(seedONS)
Q, R = torch.linalg.qr(torch.randn(n, nle))
ls = torch.zeros(nle, dtype=torch.float32)
for step in range(steps):
x = x_data[step]
D = calculate_jacobian_analytical(vanilla_rnn, h)
h, _ = vanilla_rnn(x, h)
Q = D @ Q
if step % ONSstep == 0 and nle > 0:
Q, R = torch.linalg.qr(Q)
ls += torch.log(torch.abs(torch.diag(R))) / ONSstep
return ls
# Model parameters
Nin = 1
nb_steps = 1024
nb_hidden = 64
Nout = 1
nle = 16 # number of Lyapunov exponents to floss
Ef = 41 # number of flossing epochs
# Initialize the RNN
vanilla_rnn = VanillaRNN(Nin, nb_hidden, Nout)
optimizer = optim.Adam(vanilla_rnn.parameters())
# Generate input data
pIn = 0.5 # input probability
x_data = [torch.tensor(np.random.rand(Nin) < pIn, dtype=torch.float32) for _ in range(nb_steps)]
# Optimization setup
losses = []
lyapunov_spectra = []
# Set up the initial plot
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
lyapunov_line, = ax1.plot([], [], "k", label="Lyapunov spectrum after flossing")
ax1.set_xlabel(r"Index $i$")
ax1.set_ylabel(r"Lyapunov Exponent $\lambda_i$ (1/step)")
ax1.legend()
loss_line, = ax2.semilogy([], [], "r", label="Loss")
ax2.set_xlabel("Epoch")
ax2.set_ylabel("Loss")
ax2.set_title("Loss over Epochs")
ax2.legend()
plt.ion() # Turn on interactive mode
plt.show()
lyapunov_spectrum_initial = calculate_lyapunov_spectrum(vanilla_rnn, x_data[len(x_data)//2:], nle).detach().cpu().numpy()
ax1.plot(lyapunov_spectrum_initial, "r", label="Lyapunov spectrum before flossing")
ax1.legend()
# Training loop
for epoch in range(Ef):
torch.manual_seed(epoch)
x_data = [torch.tensor(np.random.rand(Nin) < pIn, dtype=torch.float32) for _ in range(nb_steps)]
optimizer.zero_grad()
lyapunov_spectrum = calculate_lyapunov_spectrum(vanilla_rnn, x_data[len(x_data) // 2:], nle)
# Calculate the loss (mean square of the first nle Lyapunov exponents)
loss = torch.mean(lyapunov_spectrum**2)
print(f"Epoch {epoch}: Loss = {loss.item()}")
# Backward pass: compute gradients
loss.backward()
# Optimization step
optimizer.step()
# Store the loss and Lyapunov spectrum
losses.append(loss.item())
lyapunov_spectra.append(lyapunov_spectrum.detach().cpu().numpy())
if epoch % 10 == 0:
print(f"Epoch {epoch}: Loss = {loss.item()}")
# Update the Lyapunov spectrum plot
lyapunov_line.set_ydata(lyapunov_spectrum.detach().cpu().numpy())
lyapunov_line.set_xdata(range(len(lyapunov_spectrum)))
ax1.relim()
ax1.autoscale_view()
ax1.legend()
# Update the loss plot
loss_line.set_ydata(losses)
loss_line.set_xdata(range(len(losses)))
ax2.relim()
ax2.autoscale_view()
# Draw the updated plots
fig.canvas.draw()
fig.canvas.flush_events()
print("Flossing complete.")
|
