<aside> <img src="/icons/condense_yellow.svg" alt="/icons/condense_yellow.svg" width="40px" /> Python | 组卷积 | 循环组 | 可视化 | 卷积核 | 泛化 | 紧群 | 表征 | 调和分析 | 不可约 | 傅里叶 | 深度概率 | 模型 | 高维 | 结构化 | 单变量 | 分类响应 | 离散 | 潜变量 | 伯努利 | 分布 | 流生成 | 微分方程 | 神经网络
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🎯组卷积网络：实现循环组，可视化组动作，实现提升卷积核，MNIST 训练数据集训练组卷积网络的泛化能力 | 🎯可操控卷积网络：紧群的表征与调和分析，代码验证常规表征结果，不可约表征实现，傅里叶变换对群调和分析，实现可操控卷积网络 | 🎯深度概率模型：给定高维和结构化对单变量响应变量建模，实现分类响应模型，顺序响应模型、序列标记模型 | 🎯深度离散潜变量模型：使用FashionMNIST数据集，实现伯努利分布的乘积，实现均匀分类分布，测试先验分布，实现条件概率分布 | 🎯流生成模型 | 🎯超参数调优和多GPU编程 | 🎯贝叶斯神经网络实现 | 🎯非线性动力系统稀疏辨识算法 | 🎯偏微分方程神经网络求解器 | 🎯常微分方程神经网络求解器
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🎯非线性系统：Julia和Python蛛网图轨道图庞加莱截面曲面确定性非线性系统
方块，有时也被称为四块、方块，骨牌或四格，是所有已知的俄罗斯方块游戏中使用的方块。它们有七种形状，都可以旋转然后放下。方块的面积都是四个方格。在某些俄罗斯方块游戏中，它们的颜色会有所不同。四格骨牌是由四个方块组成的多格骨牌。七个单面四格骨牌分别是 I、O、T、S、Z、J 和 L。
💦使用门实现模型拟合俄罗斯方块数据集
import logging
import torch
from torch_cluster import radius_graph
from torch_geometric.data import Data, DataLoader
from torch_scatter import scatter
def tblock():
pos = [
[(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, 1, 0)],
[(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, -1, 0)],
[(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3)],
[(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 1)],
[(0, 0, 0), (1, 0, 0), (1, 1, 0), (2, 1, 0)],
]
pos = torch.tensor(pos, dtype=torch.get_default_dtype())
labels = torch.tensor(
[
[+1, 0, 0, 0, 0, 0, 0],
[-1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1],
],
dtype=torch.get_default_dtype(),
)
def mean_std(name, x) -> None:
print(f"{name} \\t{x.mean():.1f} ± ({x.var(0).mean().sqrt():.1f}|{x.std():.1f})")
class Convolution(torch.nn.Module):
def __init__(self, irreps_in, irreps_sh, irreps_out, num_neighbors) -> None:
super().__init__()
self.num_neighbors = num_neighbors
tp = FullyConnectedTensorProduct(
irreps_in1=irreps_in,
irreps_in2=irreps_sh,
irreps_out=irreps_out,
internal_weights=False,
shared_weights=False,
)
self.fc = FullyConnectedNet([3, 256, tp.weight_numel], torch.relu)
self.tp = tp
self.irreps_out = self.tp.irreps_out
def forward(self, node_features, edge_src, edge_dst, edge_attr, edge_scalars) -> torch.Tensor:
weight = self.fc(edge_scalars)
edge_features = self.tp(node_features[edge_src], edge_attr, weight)
node_features = scatter(edge_features, edge_dst, dim=0).div(self.num_neighbors**0.5)
return node_features
class Network(torch.nn.Module):
def __init__(self) -> None:
super().__init__()
self.num_neighbors = 3.8
self.irreps_sh = o3.Irreps.spherical_harmonics(3)
irreps = self.irreps_sh
# First layer with gate
gate = Gate(
"16x0e + 16x0o",
[torch.relu, torch.abs],
"8x0e + 8x0o + 8x0e + 8x0o",
[torch.relu, torch.tanh, torch.relu, torch.tanh], # gates (scalars)
"16x1o + 16x1e",
)
self.conv = Convolution(irreps, self.irreps_sh, gate.irreps_in, self.num_neighbors)
self.gate = gate
irreps = self.gate.irreps_out
# Final layer
self.final = Convolution(irreps, self.irreps_sh, "0o + 6x0e", self.num_neighbors)
self.irreps_out = self.final.irreps_out
💦多项式拟合俄罗斯方块数据集
import logging
import torch
from torch_cluster import radius_graph
from torch_geometric.data import Data, DataLoader
from torch_scatter import scatter
def tblock():
pos = [
[(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, 1, 0)],
[(0, 0, 0), (0, 0, 1), (1, 0, 0), (1, -1, 0)],
[(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3)],
[(0, 0, 0), (0, 0, 1), (0, 1, 0), (1, 0, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0)],
[(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 1)],
[(0, 0, 0), (1, 0, 0), (1, 1, 0), (2, 1, 0)],
]
pos = torch.tensor(pos, dtype=torch.get_default_dtype())
labels = torch.tensor(
[
[+1, 0, 0, 0, 0, 0, 0],
[-1, 0, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 1],
],
dtype=torch.get_default_dtype(),
)
pos = torch.einsum("zij,zaj->zai", o3.rand_matrix(len(pos)), pos)
dataset = [Data(pos=pos) for pos in pos]
data = next(iter(DataLoader(dataset, batch_size=len(dataset))))
return data, labels
class InvariantPolynomial(torch.nn.Module):
def __init__(self) -> None:
super().__init__()
self.irreps_sh: o3.Irreps = o3.Irreps.spherical_harmonics(3)
irreps_mid = o3.Irreps("64x0e + 24x1e + 24x1o + 16x2e + 16x2o")
irreps_out = o3.Irreps("0o + 6x0e")
self.tp1 = FullyConnectedTensorProduct(
irreps_in1=self.irreps_sh,
irreps_in2=self.irreps_sh,
irreps_out=irreps_mid,
)
self.tp2 = FullyConnectedTensorProduct(
irreps_in1=irreps_mid,
irreps_in2=self.irreps_sh,
irreps_out=irreps_out,
)
self.irreps_out = self.tp2.irreps_out
def forward(self, data) -> torch.Tensor:
num_neighbors = 2
num_nodes = 4 # typical number of nodes
edge_src, edge_dst = radius_graph(x=data.pos, r=1.1, batch=data.batch)
edge_vec = data.pos[edge_src] - data.pos[edge_dst]
edge_sh = o3.spherical_harmonics(
l=self.irreps_sh,
x=edge_vec,
normalize=False,
normalization="component",
)
node_features = scatter(edge_sh, edge_dst, dim=0).div(num_neighbors**0.5)
edge_features = self.tp1(node_features[edge_src], edge_sh)
node_features = scatter(edge_features, edge_dst, dim=0).div(num_neighbors**0.5)
edge_features = self.tp2(node_features[edge_src], edge_sh)
node_features = scatter(edge_features, edge_dst, dim=0).div(num_neighbors**0.5)
return scatter(node_features, data.batch, dim=0).div(num_nodes**0.5)
def main() -> None:
data, labels = tetris()
f = InvariantPolynomial()
optim = torch.optim.Adam(f.parameters(), lr=1e-2)
# == Train ==
for step in range(200):
pred = f(data)
loss = (pred - labels).pow(2).sum()
optim.zero_grad()
loss.backward()
optim.step()
if step % 10 == 0:
accuracy = pred.round().eq(labels).all(dim=1).double().mean(dim=0).item()
print(f"epoch {step:5d} | loss {loss:<10.1f} | {100 * accuracy:5.1f}% accuracy")
def test() -> None:
data, labels = tetris()
f = InvariantPolynomial()
pred = f(data)
loss = (pred - labels).pow(2).sum()
loss.backward()
rotated_data, _ = tetris()
error = f(rotated_data) - f(data)
assert error.abs().max() < 1e-5
https://embed.notionlytics.com/wt/ZXlKM2IzSnJjM0JoWTJWVWNtRmphMlZ5U1dRaU9pSlhiRWhvWlV4VVQxbHNjMlZYV2tKbU9URndaU0lzSW5CaFoyVkpaQ0k2SWpjek9ESmtOVFJoTnpCaU5qUXpOREpoTVRRMFpqSXlaV0V4TldRM05tSTVJbjA9