使用多层神经网络模型来实现图像识别功能。我们将使用 PHP 实现简化版本的神经网络，并利用它来识别手写数字图像。

<?php
// 基于多层神经网络的图像识别系统

// 定义多层神经网络类
class NeuralNetwork {
    private $inputLayerSize;
    private $hiddenLayerSize;
    private $outputLayerSize;
    private $weightsInputHidden;
    private $weightsHiddenOutput;

    public function __construct($inputLayerSize, $hiddenLayerSize, $outputLayerSize) {
        $this->inputLayerSize = $inputLayerSize;
        $this->hiddenLayerSize = $hiddenLayerSize;
        $this->outputLayerSize = $outputLayerSize;

        // 初始化权重
        $this->weightsInputHidden = $this->initializeWeights($this->inputLayerSize, $this->hiddenLayerSize);
        $this->weightsHiddenOutput = $this->initializeWeights($this->hiddenLayerSize, $this->outputLayerSize);
    }

    private function initializeWeights($inputSize, $outputSize) {
        $weights = [];
        for ($i = 0; $i < $inputSize; $i++) {
            $weights[$i] = [];
            for ($j = 0; $j < $outputSize; $j++) {
                $weights[$i][$j] = rand(-100, 100) / 100.0; // 随机初始化权重
            }
        }
        return $weights;
    }

    private function sigmoid($x) {
        return 1 / (1 + exp(-$x));
    }

    public function feedforward($inputs) {
        // 输入层到
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