AI Agent系列（10）：跨模态认知融合与多尺度能量优化

AI Agent系列（10）：跨模态认知融合与多尺度能量优化

一、异构模态纠缠架构

1. 量子神经场同步算子

import jax.numpy as jnp
from jax import grad, vmap

class QuantumHarmony:
    def __init__(self, num_modalities=5):
        self.coherence_field = jnp.zeros((num_modalities, 256))
        # 模态间耦合参数（类似杨-米尔斯场）
        self.gauge_weights = jnp.diag(jnp.linspace(0.5, 1.5, num_modalities))

    def sync_dynamics(self, modalities):
        """用规范场理论实现模态同步"""
        interaction = jnp.einsum('ma,ab,mb->m', modalities, self.gauge_weights, modalities)
        phase_flow = grad(lambda x: jnp.sum(x**3 - x))(interaction)
        return modalities + 0.1 * jnp.fft.irfft(phase_flow * jnp.fft.rfft(modalities))

class EntangledProjection:
    def __init__(self):
        self.stiefel_manifold = jnp.eye(8)[:,:3]  # 三维主干特征空间
  
    def schmidt_decomp(self, tensor):
        """施密特分解实现跨模态信息蒸馏"""
        core, factors = jnp.linalg.tensordot(tensor, self.stiefel_manifold, axes=([1],[0]))
        return jnp.einsum('ijk,kl->ijl', core, factors.T)  # 保结构投影

2. 多模态压缩流形定理

令跨模态联合分布$P(X_1,...,X_N)$在$\mathcal{M}\subset {⨂}_i{\mathbb{H}}_i^d$上满足：
$${\int }_{\mathcal{M}}{e}^{-\beta E(x)}dx=\prod _{i=1}^N{Z}_i(\beta {)}^{c_i}$$
其中$c_i$为各模态信息容量系数，$\beta$为逆认知温度参数

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二、认知能量泛函极值化

1. 变分玻尔兹曼机

import tensorflow_probability as tfp
tfd = tfp.distributions

class CognitiveEnergy(tf.Module):
    def __init__(self, num_spins=512):
        self.J = tf.Variable(tf.random.normal([num_spins, num_spins], 0, 0.02))
        self.h = tf.Variable(tf.zeros(num_spins))
  
    def free_energy(self, states):
        """计算自由能泛函（含拓扑约束项）"""
        energy = -0.5 * tf.einsum('bi,ij,bj->b', states, self.J, states) - tf.einsum('bi,i->b', states, self.h)
        entropy = tf.reduce_sum(states * tf.math.log(states + 1e-8), axis=1)
        return energy - 0.5 * entropy  # Tsallis熵修正项

class SpinGlassOptimizer(tf.optimizers.Optimizer):
    def __init__(self, learning_rate=0.01):
        super().__init__()
        self._lr = learning_rate
      
    def update_rule(self, grad, state): 
        return -self._lr * grad * (1 - state**2)  # SDE驱动更新

2. 朗之万认知动力学

随机微分方程：
$$d{\mathbf{x}}_t=-\nabla V(\mathbf{x})dt+\sqrt{2\tau }d{\mathbf{W}}_t+\alpha \sum w_{ij}{x}_j\circ d{N}_t$$
其中$d{N}_t$为泊松过程的创新噪声，$\circ$表示Stratonovich积分

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三、全息记忆重构图式

1. 记忆结晶梯度流

class MemoryCrystallization:
    def __init__(self, num_memories=1024):
        self.memory_tensor = torch.nn.Parameter(torch.randn(num_memories, 3, 256))
        self.hodge_star = torch.zeros(3,3).uniform_(-0.1,0.1)  # 外代数运算符
  
    def recall_operator(self, query):
        """基于德拉姆上同调的搜索机制"""
        differential = query @ self.memory_tensor.transpose(1,2)
        cohomology = torch.einsum('ijk,bjk->bi', self.hodge_star, differential)
        return torch.softmax(cohomology.norm(dim=1), dim=0)  # 陈类激活

class EntropyProjector:
    def __init__(self):
        self.rieffel_proj = torch.nn.Linear(1024, 256, bias=False)
      
    def stabilize(self, memory_trace):
        """Rieffel型非交换投影"""
        return self.rieffel_proj(memory_trace) @ self.rieffel_proj.weight.T

2. 记忆重整化群方程

$$\frac{\partial {\Gamma }^{(n)}}{\partial \log \Lambda }=(d-\varphi \cdot \frac{\partial }{\partial \varphi }){\Gamma }^{(n)}+\sum T_{ijk}{\Gamma }^{(i)}{\Gamma }^{(j)}{\Gamma }^{(k)}$$
其中$\Lambda$为认知截断标度，$T_{ijk}$为三重关联张量

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多尺度优化的五维约束：

1. 局域序参量守恒：$\nabla \cdot \mathbf{O}+\frac{\partial \rho }{\partial t}=S(\Sigma )$
2. 杨-米尔斯规范不变性：$\int [\mathcal{D}A]e^{-S[A]}\prod \delta (G(A))$
3. 热力学第四定律：$dE=TdS+\mu dN-pdV+\xi d\Xi$ （$\Xi$为认知功）
4. 量子纠错条件：$⟨\psi |{\mathcal{E}}^{†}(E_i{E}_j)|\psi ⟩={\delta }_{ij}$
5. 分数维度嵌入：${\dim }_{\mathbb{H}}({\mathcal{M}}_c)=\frac{\log N(ϵ)}{\log (1/ϵ)}$