17000.机器学习-数字1-9实例

guangshui5162025-01-01 15:20

文章目录

  • [1 搭建环境](#1 搭建环境)
    • [1.1 下载 libtorch c++ 版本](#1.1 下载 libtorch c++ 版本)
    • [1.2 下载数字数据集](#1.2 下载数字数据集)
    • [1.3 目录结构](#1.3 目录结构)
  • [2 源码目录](#2 源码目录)
    • [2.1 CMakeLists.txt文件](#2.1 CMakeLists.txt文件)
    • [2.2 训练集主程序 main.cpp](#2.2 训练集主程序 main.cpp)
    • [2.2 dataset.h](#2.2 dataset.h)
    • [2.3 model.cpp](#2.3 model.cpp)
    • [2.4 识别主程序](#2.4 识别主程序)
  • [3 运行结果](#3 运行结果)
    • [3.1 训练集程序运行](#3.1 训练集程序运行)
    • [3.2 识别程序](#3.2 识别程序)

1 搭建环境

1.1 下载 libtorch c++ 版本

由于我当前采用虚拟机形式，使用cpu处理器，没有使用GPU处理器。

cpp 复制代码 
wget https://download.pytorch.org/libtorch/cpu/libtorch-cxx11-abi-shared-with-deps-2.0.0%2Bcpu.zip
unzip libtorch-cxx11-abi-shared-with-deps-2.0.0+cpu.zip

1.2 下载数字数据集

c 复制代码 
wget https://ossci-datasets.s3.amazonaws.com/mnist/train-images-idx3-ubyte.gz
wget https://ossci-datasets.s3.amazonaws.com/mnist/train-labels-idx1-ubyte.gz
wget https://ossci-datasets.s3.amazonaws.com/mnist/t10k-images-idx3-ubyte.gz
wget https://ossci-datasets.s3.amazonaws.com/mnist/t10k-labels-idx1-ubyte.gz

 gunzip t10k-images-idx3-ubyte.gz
 gunzip t10k-labels-idx1-ubyte.gz
 gunzip train-images-idx3-ubyte.gz
 gunzip train-labels-idx1-ubyte.gz

1.3 目录结构

bash 复制代码 
xhome@ubuntu:~/project_ai$ ls
data  image  libtorch  mnist_demo

2 源码目录

bash 复制代码 
xhome@ubuntu:~/project_ai/mnist_demo$ ls
build  CMakeLists.txt  src

xhome@ubuntu:~/project_ai/mnist_demo/src$ ls
dataset.h  main.cpp  model.cpp  model.h  predict.cpp

2.1 CMakeLists.txt文件

xhome@ubuntu:~/project_ai/mnist_demo$ cat CMakeLists.txt

cpp 复制代码 
cmake_minimum_required(VERSION 3.0)
project(mnist_demo)

set(CMAKE_CXX_STANDARD 14)
set(CMAKE_CXX_STANDARD_REQUIRED ON)

# 直接在这里设置 LibTorch 路径（修改为你的实际路径）
#set(CMAKE_PREFIX_PATH "${CMAKE_CURRENT_SOURCE_DIR}/libtorch")
# 或者使用绝对路径
set(CMAKE_PREFIX_PATH "/home/xhome/project_ai/libtorch")

find_package(Torch REQUIRED)
find_package(OpenCV REQUIRED)

# 添加源文件
add_executable(mnist_demo 
    src/main.cpp
    src/model.cpp
)

# 添加预测程序
add_executable(mnist_predict
    src/predict.cpp
    src/model.cpp
)

# 包含头文件目录
target_include_directories(mnist_demo PRIVATE ${CMAKE_CURRENT_SOURCE_DIR}/src)

# 链接 LibTorch
target_link_libraries(mnist_demo "${TORCH_LIBRARIES}" ${OpenCV_LIBS} pthread)
target_link_libraries(mnist_predict ${TORCH_LIBRARIES} ${OpenCV_LIBS} pthread)

# 设置编译选项
set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${TORCH_CXX_FLAGS}")

2.2 训练集主程序 main.cpp

xhome@ubuntu:~/project_ai/mnist_demo/src$ cat main.cpp

cpp 复制代码 
#include <torch/torch.h>
#include "model.h"
#include "dataset.h"
#include <iostream>
#include <iomanip>
#include <chrono>


// 训练一个epoch
template<typename DataLoader>
void train(
    Net& model,
    DataLoader& data_loader,
    torch::optim::Optimizer& optimizer,
    size_t epoch,
    size_t dataset_size) {
    
    model.train();
    size_t batch_idx = 0;
    size_t processed = 0;
    float last_loss = 0.0f;  // 添加这行来存储最后一个loss值
    
    for (auto& batch : data_loader) {
        auto data = batch.data;
        auto target = batch.target;

        optimizer.zero_grad();
        auto output = model.forward(data);
        auto loss = torch::nll_loss(output, target);
        
        loss.backward();
        optimizer.step();

        processed += batch.data.size(0);
        last_loss = loss.template item<float>();  // 保存最后一个loss值
        
        if (batch_idx++ % 100 == 0) {
            std::cout << "\rTrain Epoch: " << epoch 
                      << " [" << processed << "/" 
                      << dataset_size
                      << " (" << std::fixed << std::setprecision(1)
                      << (100.0 * processed / dataset_size)
                      << "%)] Loss: " << std::fixed << std::setprecision(4)
                      << last_loss << std::flush;
        }
    }
    
    // 在epoch结束时显示最终进度
    std::cout << "\rTrain Epoch: " << epoch 
              << " [" << dataset_size << "/" 
              << dataset_size
              << " (100.0%)] Loss: " << std::fixed << std::setprecision(4)
              << last_loss << std::endl;
}

// 测试模型
template<typename DataLoader>
void test(
    Net& model,
    DataLoader& data_loader,
    size_t dataset_size) {
    
    model.eval();
    double test_loss = 0;
    int32_t correct = 0;
    
    torch::NoGradGuard no_grad;
    
    for (const auto& batch : data_loader) {
        auto data = batch.data;
        auto target = batch.target;
        
        auto output = model.forward(data);
        test_loss += torch::nll_loss(
            output, 
            target, 
            /*weight=*/{},
            torch::Reduction::Sum
        ).template item<float>();
        
        auto pred = output.argmax(1);
        correct += pred.eq(target).sum().template item<int64_t>();
    }

    test_loss /= dataset_size;
    double accuracy = 100.0 * correct / dataset_size;

    std::cout << "Test set: Average loss: " << std::fixed << std::setprecision(4)
              << test_loss << ", Accuracy: " << correct << "/"
              << dataset_size << " (" << std::fixed << std::setprecision(1)
              << accuracy << "%)\n";
}

int main() {
    try {
        // 设置随机种子
        torch::manual_seed(1);
        
        // 使用CPU
        torch::Device device(torch::kCPU);
        std::cout << "Training on CPU." << std::endl;

        // 创建模型
        auto model = std::make_shared<Net>();
        model->to(device);

         // 创建数据集
        const std::string data_path = "data/MNIST/raw";
        
        std::cout << "Loading training dataset..." << std::endl;
        auto train_dataset = MNISTDataset(data_path, MNISTDataset::Mode::kTrain)
            .map(torch::data::transforms::Stack<>());
            
        std::cout << "Loading test dataset..." << std::endl;
        auto test_dataset = MNISTDataset(data_path, MNISTDataset::Mode::kTest)
            .map(torch::data::transforms::Stack<>());

        // 获取并存储数据集大小
        const size_t train_size = train_dataset.size().value();
        const size_t test_size = test_dataset.size().value();
        
        std::cout << "Training dataset size: " << train_size << std::endl;
        std::cout << "Test dataset size: " << test_size << std::endl;

        // 创建数据加载器
        auto train_loader = torch::data::make_data_loader<torch::data::samplers::RandomSampler>(
            std::move(train_dataset),
            torch::data::DataLoaderOptions().batch_size(32).workers(1));
            
        auto test_loader = torch::data::make_data_loader(
            std::move(test_dataset),
            torch::data::DataLoaderOptions().batch_size(100).workers(1));
        
        // 创建优化器
        torch::optim::SGD optimizer(
            model->parameters(), 
            torch::optim::SGDOptions(0.01).momentum(0.5));

        // 记录开始时间
        auto start_time = std::chrono::high_resolution_clock::now();

        // 训练循环
        const int num_epochs = 10;
        for (size_t epoch = 1; epoch <= num_epochs; ++epoch) {
            std::cout << "\nEpoch " << epoch << "/" << num_epochs << std::endl;
            
            train(*model, *train_loader, optimizer, epoch, train_size);
            test(*model, *test_loader, test_size);
        }

        // 计算总训练时间
        auto end_time = std::chrono::high_resolution_clock::now();
        auto duration = std::chrono::duration_cast<std::chrono::minutes>(
            end_time - start_time);
        
        std::cout << "\nTraining completed in " 
                  << duration.count() << " minutes" << std::endl;

        // 保存模型
        torch::save(model, "mnist_cnn.pt");
        std::cout << "Model saved to mnist_cnn.pt" << std::endl;
        
    } catch (const std::exception& e) {
        std::cerr << "Error: " << e.what() << std::endl;
        return -1;
    }
    
    return 0;
}

2.2 dataset.h

cpp 复制代码 
#ifndef __DATA_SET_HEAD_H
#define __DATA_SET_HEAD_H

#include <torch/torch.h>
#include <string>
#include <vector>
#include <fstream>
#include <iostream>

class MNISTDataset : public torch::data::Dataset<MNISTDataset> {
public:
    enum Mode { kTrain, kTest };
    
    explicit MNISTDataset(const std::string& root, Mode mode = Mode::kTrain) {
        std::string prefix = mode == Mode::kTrain ? "train" : "t10k";
        
        // 读取图像
        std::string image_file = root + "/" + prefix + "-images-idx3-ubyte";
        std::cout << "Loading images from: " << image_file << std::endl;
        images = read_images(image_file);
        
        // 读取标签
        std::string label_file = root + "/" + prefix + "-labels-idx1-ubyte";
        std::cout << "Loading labels from: " << label_file << std::endl;
        labels = read_labels(label_file);
        
        std::cout << "Dataset size: " << images.size(0) << std::endl;
    }
    
    torch::data::Example<> get(size_t index) override {
        return {images[index], labels[index]};
    }
    
    torch::optional<size_t> size() const override {
        return images.size(0);
    }

private:
    torch::Tensor images, labels;
    
    torch::Tensor read_images(const std::string& path) {
        std::ifstream file(path, std::ios::binary);
        if (!file) {
            throw std::runtime_error("Cannot open file: " + path);
        }
        
        int32_t magic_number = 0, n_images = 0, n_rows = 0, n_cols = 0;
        
        file.read(reinterpret_cast<char*>(&magic_number), sizeof(magic_number));
        file.read(reinterpret_cast<char*>(&n_images), sizeof(n_images));
        file.read(reinterpret_cast<char*>(&n_rows), sizeof(n_rows));
        file.read(reinterpret_cast<char*>(&n_cols), sizeof(n_cols));
        
        // 转换字节序
        magic_number = __builtin_bswap32(magic_number);
        n_images = __builtin_bswap32(n_images);
        n_rows = __builtin_bswap32(n_rows);
        n_cols = __builtin_bswap32(n_cols);
        
        // 读取图像数据
        std::vector<uint8_t> buffer(n_images * n_rows * n_cols);
        file.read(reinterpret_cast<char*>(buffer.data()), buffer.size());
        
        auto tensor = torch::from_blob(buffer.data(), 
            {n_images, n_rows, n_cols}, torch::kUInt8).clone();
        
        // 添加通道维度并归一化
        tensor = tensor.unsqueeze(1).to(torch::kFloat32).div(255.0);
        return tensor;
    }
    
    torch::Tensor read_labels(const std::string& path) {
        std::ifstream file(path, std::ios::binary);
        if (!file) {
            throw std::runtime_error("Cannot open file: " + path);
        }
        
        int32_t magic_number = 0, n_labels = 0;
        
        file.read(reinterpret_cast<char*>(&magic_number), sizeof(magic_number));
        file.read(reinterpret_cast<char*>(&n_labels), sizeof(n_labels));
        
        magic_number = __builtin_bswap32(magic_number);
        n_labels = __builtin_bswap32(n_labels);
        
        std::vector<uint8_t> buffer(n_labels);
        file.read(reinterpret_cast<char*>(buffer.data()), n_labels);
        
        return torch::from_blob(buffer.data(), 
            {n_labels}, torch::kUInt8).clone().to(torch::kLong);
    }
};


#endif

2.3 model.cpp

cpp 复制代码 
#include "model.h"

Net::Net() {
    // 第一个卷积块: 1->32 通道, 3x3 卷积核
    conv1 = register_module("conv1", 
        torch::nn::Conv2d(
            torch::nn::Conv2dOptions(1, 32, 3)
                .stride(1)
                .padding(1)
        ));
    batch_norm1 = register_module("batch_norm1",
        torch::nn::BatchNorm2d(32));
    
    // 第二个卷积块: 32->64 通道, 3x3 卷积核
    conv2 = register_module("conv2", 
        torch::nn::Conv2d(
            torch::nn::Conv2dOptions(32, 64, 3)
                .stride(1)
                .padding(1)
        ));
    batch_norm2 = register_module("batch_norm2",
        torch::nn::BatchNorm2d(64));
    
    // 全连接层
    fc1 = register_module("fc1", 
        torch::nn::Linear(7 * 7 * 64, 128));
    fc2 = register_module("fc2", 
        torch::nn::Linear(128, 10));
    
    // Dropout层
    dropout = register_module("dropout",
        torch::nn::Dropout(torch::nn::DropoutOptions(0.25)));
}

torch::Tensor Net::forward(torch::Tensor x) {
    // 第一个卷积块
    x = conv1->forward(x);
    x = batch_norm1->forward(x);
    x = torch::relu(x);
    x = torch::max_pool2d(x, 2);
    
    // 第二个卷积块
    x = conv2->forward(x);
    x = batch_norm2->forward(x);
    x = torch::relu(x);
    x = torch::max_pool2d(x, 2);
    
    // 展平
    x = x.view({-1, 7 * 7 * 64});
    
    // 全连接层
    x = torch::relu(fc1->forward(x));
    x = dropout->forward(x);
    x = fc2->forward(x);
    
    return torch::log_softmax(x, 1);
}

2.4 识别主程序

cpp 复制代码 
#include <torch/torch.h>
#include <opencv2/opencv.hpp>
#include <iostream>
#include <iomanip>
#include <memory>
#include <string>
#include "model.h"

torch::Tensor preprocess_image(const std::string& image_path) {
    // 读取图片
    cv::Mat image = cv::imread(image_path, cv::IMREAD_GRAYSCALE);
    if (image.empty()) {
        throw std::runtime_error("Error: Could not read image: " + image_path);
    }

    std::cout << "Original image size: " << image.size() << std::endl;

    // 使用Otsu's方法进行自动阈值二值化
    cv::Mat binary;
    cv::threshold(image, binary, 0, 255, cv::THRESH_BINARY | cv::THRESH_OTSU);

    // 应用形态学操作来改善数字形状
    cv::Mat morph;
    cv::Mat kernel = cv::getStructuringElement(cv::MORPH_ELLIPSE, cv::Size(3, 3));
    cv::morphologyEx(binary, morph, cv::MORPH_CLOSE, kernel);

    // 找到数字的边界框
    std::vector<std::vector<cv::Point>> contours;
    cv::findContours(255 - morph, contours, cv::RETR_EXTERNAL, cv::CHAIN_APPROX_SIMPLE);
    
    // 找到最大的轮廓
    cv::Rect boundingBox;
    double maxArea = 0;
    for (const auto& contour : contours) {
        double area = cv::contourArea(contour);
        if (area > maxArea) {
            maxArea = area;
            boundingBox = cv::boundingRect(contour);
        }
    }

    // 确保边界框是正方形，并且保持纵横比
    int maxSide = std::max(boundingBox.width, boundingBox.height);
    int xCenter = boundingBox.x + boundingBox.width / 2;
    int yCenter = boundingBox.y + boundingBox.height / 2;
    
    // 扩展边界框为正方形
    boundingBox.x = xCenter - maxSide / 2;
    boundingBox.y = yCenter - maxSide / 2;
    boundingBox.width = maxSide;
    boundingBox.height = maxSide;

    // 添加padding
    int padding = maxSide / 4;
    boundingBox.x = std::max(0, boundingBox.x - padding);
    boundingBox.y = std::max(0, boundingBox.y - padding);
    boundingBox.width = std::min(image.cols - boundingBox.x, boundingBox.width + 2 * padding);
    boundingBox.height = std::min(image.rows - boundingBox.y, boundingBox.height + 2 * padding);

    // 裁剪图像
    cv::Mat cropped = morph(boundingBox);

    // 调整大小为20x20
    cv::Mat resized;
    cv::resize(cropped, resized, cv::Size(20, 20), 0, 0, cv::INTER_AREA);

    // 添加4像素边框
    cv::Mat padded;
    cv::copyMakeBorder(resized, padded, 4, 4, 4, 4, 
                      cv::BORDER_CONSTANT, cv::Scalar(255));

    // 反转颜色
    cv::Mat inverted;
    cv::bitwise_not(padded, inverted);

    // 应用轻微的高斯模糊
    cv::Mat blurred;
    cv::GaussianBlur(inverted, blurred, cv::Size(3, 3), 0.5);

    // 再次应用阈值，确保清晰的边界
    cv::Mat final;
    cv::threshold(blurred, final, 127, 255, cv::THRESH_BINARY);

    // 转换为浮点数并归一化
    cv::Mat float_img;
    final.convertTo(float_img, CV_32F, 1.0/255.0);

    // 保存所有处理步骤的图片
    cv::imwrite("step1_binary.jpg", binary);
    cv::imwrite("step2_morph.jpg", morph);
    cv::imwrite("step3_cropped.jpg", cropped);
    cv::imwrite("step4_resized.jpg", resized);
    cv::imwrite("step5_padded.jpg", padded);
    cv::imwrite("step6_inverted.jpg", inverted);
    cv::imwrite("step7_final.jpg", final);
    
    std::cout << "Preprocessing steps saved as images" << std::endl;

    // 转换为tensor
    auto tensor = torch::from_blob(float_img.data, {1, 28, 28}, torch::kFloat32).clone();
    tensor = tensor.unsqueeze(0);

    return tensor;
}



void display_tensor(const torch::Tensor& tensor) {
    std::cout << "\nProcessed image (ASCII art):\n";
    for (int i = 0; i < 28; ++i) {
        for (int j = 0; j < 28; ++j) {
            float pixel = tensor[0][0][i][j].item<float>();
            if (pixel < 0.2) std::cout << "  ";
            else if (pixel < 0.4) std::cout << "..";
            else if (pixel < 0.6) std::cout << "**";
            else if (pixel < 0.8) std::cout << "##";
            else std::cout << "@@";
        }
        std::cout << std::endl;
    }
}

int main(int argc, char* argv[]) {
    if (argc < 2) {
        std::cerr << "Usage: " << argv[0] << " <image_path>" << std::endl;
        return 1;
    }

    try {
        // 使用CPU
        torch::Device device(torch::kCPU);
        
        // 加载模型
        auto model = std::make_shared<Net>();
        torch::load(model, "mnist_cnn.pt");
        model->to(device);
        model->eval();

        // 预处理图片
        auto input = preprocess_image(argv[1]);
        
        // 显示处理后的图像
        display_tensor(input);

        // 进行预测
        torch::NoGradGuard no_grad;
        auto output = model->forward(input);
        auto probabilities = torch::softmax(output, 1);
        auto prediction = output.argmax(1);

        // 打印预测结果
        std::cout << "\nPredicted digit: " << prediction.item<int>() << std::endl;

        // 打印每个数字的概率
        std::cout << "\nProbabilities for each digit:" << std::endl;
        for (int i = 0; i < 10; ++i) {
            std::cout << "Digit " << i << ": " 
                      << std::fixed << std::setprecision(4) 
                      << probabilities[0][i].item<float>() * 100 << "%" 
                      << std::endl;
        }

    } catch (const std::exception& e) {
        std::cerr << "Error: " << e.what() << std::endl;
        return -1;
    }
    
    return 0;
}

3 运行结果

3.1 训练集程序运行

cpp 复制代码 
xhome@ubuntu:~/project_ai/mnist_demo/build$ ./mnist_demo 
Training on CPU.
Loading training dataset...
Loading images from: data/MNIST/raw/train-images-idx3-ubyte
Loading labels from: data/MNIST/raw/train-labels-idx1-ubyte
Dataset size: 60000
Loading test dataset...
Loading images from: data/MNIST/raw/t10k-images-idx3-ubyte
Loading labels from: data/MNIST/raw/t10k-labels-idx1-ubyte
Dataset size: 10000
Training dataset size: 60000
Test dataset size: 10000

Epoch 1/10
Train Epoch: 1 [60000/60000 (100.0%)] Loss: 0.0716
Test set: Average loss: 0.0504, Accuracy: 9840/10000 (98.4%)

Epoch 2/10
Train Epoch: 2 [60000/60000 (100.0%)] Loss: 0.0070
Test set: Average loss: 0.0420, Accuracy: 9863/10000 (98.6%)

Epoch 3/10
Train Epoch: 3 [60000/60000 (100.0%)] Loss: 0.0337
Test set: Average loss: 0.0352, Accuracy: 9889/10000 (98.9%)

Epoch 4/10
Train Epoch: 4 [60000/60000 (100.0%)] Loss: 0.0009
Test set: Average loss: 0.0325, Accuracy: 9893/10000 (98.9%)

Epoch 5/10
Train Epoch: 5 [60000/60000 (100.0%)] Loss: 0.0131
Test set: Average loss: 0.0277, Accuracy: 9907/10000 (99.1%)

Epoch 6/10
Train Epoch: 6 [60000/60000 (100.0%)] Loss: 0.0219
Test set: Average loss: 0.0270, Accuracy: 9910/10000 (99.1%)

Epoch 7/10
Train Epoch: 7 [60000/60000 (100.0%)] Loss: 0.0033
Test set: Average loss: 0.0254, Accuracy: 9917/10000 (99.2%)

Epoch 8/10
Train Epoch: 8 [60000/60000 (100.0%)] Loss: 0.0033
Test set: Average loss: 0.0247, Accuracy: 9915/10000 (99.2%)

Epoch 9/10
Train Epoch: 9 [60000/60000 (100.0%)] Loss: 0.0561
Test set: Average loss: 0.0221, Accuracy: 9926/10000 (99.3%)

Epoch 10/10
Train Epoch: 10 [60000/60000 (100.0%)] Loss: 0.0003
Test set: Average loss: 0.0248, Accuracy: 9914/10000 (99.1%)

Training completed in 1 minutes
Model saved to mnist_cnn.pt

3.2 识别程序

cpp 复制代码 
xhome@ubuntu:~/project_ai/mnist_demo/build$ ./mnist_predict ../../image/1.png 
Original image size: [456 x 479]
Preprocessing steps saved as images

Processed image (ASCII art):
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                        @@@@@@                          
                    @@@@@@@@@@                          
                      @@@@@@@@                          
                      @@@@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                        @@@@@@                          
                      @@@@@@@@                          
                      @@@@@@@@                          
                    @@@@@@@@@@@@                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        

Predicted digit: 1

Probabilities for each digit:
Digit 0: 0.0047%
Digit 1: 94.2735%
Digit 2: 0.0737%
Digit 3: 0.0274%
Digit 4: 0.0054%
Digit 5: 2.7657%
Digit 6: 0.0065%
Digit 7: 0.0035%
Digit 8: 2.7485%
Digit 9: 0.0911%
cpp 复制代码 
xhome@ubuntu:~/project_ai/mnist_demo/build$ ./mnist_predict ../../image/5.png 
Original image size: [508 x 512]
Preprocessing steps saved as images

Processed image (ASCII art):
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        
                    @@@@@@@@@@@@@@@@                    
                    @@@@@@@@@@@@@@@@                    
                    @@@@@@@@                            
                    @@@@@@@@                            
                    @@@@@@@@@@@@@@@@                    
                    @@@@@@@@@@@@@@@@@@                  
                    @@@@@@@@  @@@@@@@@                  
                              @@@@@@@@                  
                              @@@@@@@@                  
                    @@@@@@@@  @@@@@@@@                  
                    @@@@@@@@  @@@@@@@@                  
                    @@@@@@@@  @@@@@@@@                  
                    @@@@@@@@@@@@@@@@@@                  
                      @@@@@@@@@@@@@@                    
                          @@@@@@@@                      
                                                        
                                                        
                                                        
                                                        
                                                        
                                                        

Predicted digit: 5

Probabilities for each digit:
Digit 0: 0.0118%
Digit 1: 0.0003%
Digit 2: 0.0079%
Digit 3: 0.3717%
Digit 4: 0.0000%
Digit 5: 84.2349%
Digit 6: 0.0197%
Digit 7: 0.0005%
Digit 8: 13.8340%
Digit 9: 1.5192%
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