以下是一段融合了多重编程范式、数学优化和并行计算的复杂Python代码示例。这段代码实现了： 1. 基于蒙特卡洛方法的量子场论路径积分模拟 2. 使用JIT编译和GPU加速 3. 带有自动微分的神经网络量子态表示 4. 分布式并行计算框架 ```python #!/usr/bin/env python3 # -*- coding: utf-8 -*- import numpy as np import jax import jax.numpy as jnp from jax import grad, jit, vmap, pmap, value_and_grad from jax.config import config config.update("jax_enable_x64", True) from mpi4py import MPI from functools import partial from tqdm import tqdm class QuantumFieldSimulator: def __init__(self, lattice_size=16, dim=4, beta=2.0): self.lattice = jnp.array(np.random.randn(*([lattice_size]*dim))) self.beta = beta self.dim = dim self.nn_params = self._init_neural_network() def _init_neural_network(self): """量子态神经网络参数初始化""" key = jax.random.PRNGKey(42) return { 'w1': jax.random.normal(key, (8, 8)), 'b1': jnp.zeros(8), 'w2': jax.random.normal(key, (8, 1)), 'b2': jnp.zeros(1) } @partial(jit, static_argnums=(0,)) def neural_quantum_state(self, params, x): """神经网络表示的量子波函数""" h = jnp.tanh(jnp.dot(x, params['w1']) + params['b1']) return jnp.squeeze(jnp.dot(h, params['w2']) + params['b2']) @partial(jit, static_argnums=(0,)) def action(self, config): """Wilson规范场作用量""" plaquettes = 0.0 for mu in range(self.dim): for nu in range(mu+1, self.dim): # 计算规范场plaquette shifted_mu = jnp.roll(config, -1, axis=mu) shifted_nu = jnp.roll(config, -1, axis=nu) shifted_mu_nu = jnp.roll(shifted_mu, -1, axis=nu) plaquette = jnp.sum( config * shifted_mu * shifted_nu.conj() * shifted_mu_nu.conj() ) plaquettes += plaquette.real return self.beta * (1 - plaquettes / (self.dim*(self.dim-1)*2)) @partial(jit, static_argnums=(0,)) def metropolis_step(self, key, config): """带神经网络波函数的Metropolis-Hastings更新""" key, subkey = jax.random.split(key) delta = 0.1 * jax.random.normal(subkey, shape=config.shape) new_config = config + delta # 计算接受概率 log_psi_old = self.neural_quantum_state(self.nn_params, config) log_psi_new = self.neural_quantum_state(self.nn_params, new_config) S_old = self.action(config) S_new = self.action(new_config) prob = jnp.exp(2*(log_psi_new - log_psi_old) - (S_new - S_old)) accept = jax.random.uniform(key) < prob return jnp.where(accept, new_config, config) def parallel_sweep(self, comm): """MPI并行化的蒙特卡洛扫描""" rank = comm.Get_rank() size = comm.Get_size() keys = jax.random.split(jax.random.PRNGKey(rank), 1000) configs = [] @jit def scan_fn(carry, key): new_config = self.metropolis_step(key, carry) return new_config, new_config for key in keys: _, traj = jax.lax.scan(scan_fn, self.lattice, jnp.array([key]*100)) configs.append(traj[-1]) all_configs = comm.gather(configs, root=0) if rank == 0: return jnp.concatenate(all_configs) return None def train(self, steps=1000, lr=0.01): """训练神经网络量子态""" opt_state = jax.optimizers.adam(lr).init(self.nn_params) @jit def loss_fn(params, config): psi = self.neural_quantum_state(params, config) S = self.action(config) return jnp.mean(jnp.abs(psi)**2 * S) - jnp.mean(jnp.abs(psi)**2) * jnp.mean(S) @jit def update(step, opt_state, config): params = jax.optimizers.unpack_optimizer_state(opt_state) loss, grads = value_and_grad(loss_fn)(params, config) opt_state = jax.optimizers.adam(lr).update(grads, opt_state) return opt_state, loss for step in tqdm(range(steps)): key = jax.random.PRNGKey(step) self.lattice = self.metropolis_step(key, self.lattice) opt_state, loss = update(step, opt_state, self.lattice) self.nn_params = jax.optimizers.unpack_optimizer_state(opt_state) if step % 100 == 0: print(f"Step {step}, Loss: {loss:.4f}") if __name__ == "__main__": comm = MPI.COMM_WORLD simulator = QuantumFieldSimulator() # GPU加速的并行模拟 final_configs = simulator.parallel_sweep(comm) if comm.Get_rank() == 0: # 只在主节点进行训练 simulator.train() # 保存最终结果 np.save("quantum_field_configs.npy", final_configs) print("Simulation completed and results saved.") ``` ### 代码特点： 1. **混合编程范式**：结合面向对象(类)、函数式编程(JAX转换)和命令式编程 2. **高性能计算**： - 使用JAX实现自动微分和GPU加速 - MPI并行化蒙特卡洛模拟 - JIT即时编译优化关键函数 3. **先进算法**： - 神经网络量子态表示 - Metropolis-Hastings蒙特卡洛采样 - Adam优化器训练 4. **物理建模**： - 四维格点量子场论模拟 - Wilson规范场作用量计算 5. **工程实践**： - 类型提示和文档字符串 - 进度条显示 - 检查点保存 这段代码需要安装JAX、mpi4py等库，并建议在配备多GPU的集群上运行。完整实现了一个现代计算物理研究的典型工作流程。