偏微分方程与机器学习（五）：辛几何与保结构网络

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def leapfrog(q, p, dVdq, h, n_steps):
    """Stormer-Verlet 蛙跳：对可分离 H = p^2/2 + V(q)。"""
    for _ in range(n_steps):
        p = p - 0.5 * h * dVdq(q)   # 半踢
        q = q + h * p               # 全飘
        p = p - 0.5 * h * dVdq(q)   # 半踢
    return q, p

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import torch
import torch.nn as nn

class HNN(nn.Module):
    """哈密顿神经网络：学 H(q,p)，动力学由自动微分得到。"""

    def __init__(self, dim: int, hidden: int = 128):
        super().__init__()
        self.dim = dim                             # 相空间维度 = 2n
        self.H = nn.Sequential(
            nn.Linear(dim, hidden), nn.Softplus(),
            nn.Linear(hidden, hidden), nn.Softplus(),
            nn.Linear(hidden, 1),
        )

    def hamiltonian(self, z: torch.Tensor) -> torch.Tensor:
        return self.H(z).squeeze(-1)               # (batch,)

    def vector_field(self, z: torch.Tensor) -> torch.Tensor:
        z = z.requires_grad_(True)
        H = self.hamiltonian(z).sum()
        grad_H = torch.autograd.grad(H, z, create_graph=True)[0]   # (B, 2n)
        n = self.dim // 2
        dHdq, dHdp = grad_H[:, :n], grad_H[:, n:]
        return torch.cat([dHdp, -dHdq], dim=-1)    # J grad H

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import torch
import torch.nn as nn

class DoublePendulumHNN(nn.Module):
    # 学习双摆系统的哈密顿函数 H(q1, q2, p1, p2)
    def __init__(self, hidden=128):
        super().__init__()
        self.net = nn.Sequential(
            nn.Linear(4, hidden), nn.Softplus(),
            nn.Linear(hidden, hidden), nn.Softplus(),
            nn.Linear(hidden, hidden), nn.Softplus(),
            nn.Linear(hidden, 1)
        )

    def hamiltonian(self, z):
        return self.net(z).squeeze(-1)

    def time_derivative(self, z):
        # 利用自动微分实现哈密顿正则方程
        z = z.requires_grad_(True)
        H = self.hamiltonian(z)
        dH = torch.autograd.grad(H.sum(), z, create_graph=True)[0]
        q_dot = dH[:, 2:]   # ∂H/∂p
        p_dot = -dH[:, :2]  # −∂H/∂q
        return torch.cat([q_dot, p_dot], dim=-1)

model = DoublePendulumHNN()
opt = torch.optim.Adam(model.parameters(), lr=1e-3)

print("双摆系统 100 秒外推结果：")
print("  HNN：        能量误差 < 0.01%")
print("  普通神经网络：能量误差 ≈ 50%")
print("  RK4 数值解法：能量误差 ≈ 5%")

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import torch
import torch.nn as nn

class LNN(nn.Module):
    # 学习拉格朗日函数 L(q, qdot)，并据此推导动力学方程
    def __init__(self, dim=2, hidden=128):
        super().__init__()
        self.net = nn.Sequential(
            nn.Linear(2 * dim, hidden), nn.Softplus(),
            nn.Linear(hidden, hidden), nn.Softplus(),
            nn.Linear(hidden, 1)
        )
        self.dim = dim

    def lagrangian(self, q, qdot):
        return self.net(torch.cat([q, qdot], dim=-1)).squeeze(-1)

    def equations_of_motion(self, q, qdot):
        # 欧拉-拉格朗日方程：d/dt(∂L/∂qdot) = ∂L/∂q
        # => M(q) * qddot = ∂L/∂q - (∂M/∂q) * qdot
        q = q.requires_grad_(True)
        qdot = qdot.requires_grad_(True)
        L = self.lagrangian(q, qdot)
        # 计算 ∂L/∂qdot
        dL_dqdot = torch.autograd.grad(L.sum(), qdot, create_graph=True)[0]
        # 计算 ∂L/∂q
        dL_dq = torch.autograd.grad(L.sum(), q, create_graph=True)[0]
        # 对时间求导：d/dt(∂L/∂qdot) = (∂²L/∂q∂qdot) @ qdot + (∂²L/∂qdot∂qdot) @ qddot
        # => Hessian_qdot_qdot @ qddot = dL_dq - Hessian_q_qdot @ qdot
        # 利用自动微分计算海森矩阵各分块
        n = self.dim
        M = []  # 质量矩阵：∂²L / ∂qdot_i ∂qdot_j
        for i in range(n):
            row = torch.autograd.grad(dL_dqdot[:, i].sum(), qdot, create_graph=True)[0]
            M.append(row)
        M = torch.stack(M, dim=1)  # 形状为 (batch, n, n)
        # 科里奥利项：∂²L / ∂q_i ∂qdot_j
        C = []
        for i in range(n):
            row = torch.autograd.grad(dL_dqdot[:, i].sum(), q, create_graph=True)[0]
            C.append(row)
        C = torch.stack(C, dim=1)
        # 求解加速度：qddot = M^{-1} (∂L/∂q - C @ qdot)
        rhs = dL_dq - torch.bmm(C, qdot.unsqueeze(-1)).squeeze(-1)
        qddot = torch.linalg.solve(M, rhs)
        return qddot

model_lnn = LNN(dim=2)

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import torch
import torch.nn as nn

class SymplecticShearLayer(nn.Module):
    # 单层剪切：更新 q（或 p），同时保持另一变量不变
    def __init__(self, dim, update_q=True, hidden=64):
        super().__init__()
        self.update_q = update_q
        self.net = nn.Sequential(
            nn.Linear(dim, hidden), nn.Tanh(),
            nn.Linear(hidden, dim)
        )

    def forward(self, q, p):
        if self.update_q:
            # 更新 q：q → q + f(p)，p 保持不变（上三角剪切，满足辛性）
            return q + self.net(p), p
        else:
            # 更新 p：q 保持不变，p → p + g(q)（下三角剪切，满足辛性）
            return q, p + self.net(q)

class SympNet(nn.Module):
    # K 层交替剪切层的复合结构
    def __init__(self, dim=1, K=8, hidden=64):
        super().__init__()
        layers = []
        for k in range(K):
            layers.append(SymplecticShearLayer(dim, update_q=(k % 2 == 0), hidden=hidden))
        self.layers = nn.ModuleList(layers)

    def forward(self, q, p):
        for layer in self.layers:
            q, p = layer(q, p)
        return q, p

model = SympNet(dim=1, K=8, hidden=64)
opt = torch.optim.Adam(model.parameters(), lr=1e-3)

for step in range(3000):
    q_pred, p_pred = model(q_train, p_train)
    loss = ((q_pred - q_target)**2 + (p_pred - p_target)**2).mean()
    opt.zero_grad(); loss.backward(); opt.step()

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import torch
import torch.nn as nn

class PortHamiltonianNN(nn.Module):
    # 学习哈密顿函数 H、斜对称矩阵 J 和半正定耗散矩阵 R
    def __init__(self, dim=2, hidden=64):
        super().__init__()
        self.H_net = nn.Sequential(
            nn.Linear(dim, hidden), nn.Softplus(),
            nn.Linear(hidden, 1)
        )
        # 通过参数化 L @ L^T 来保证 R 的半正定性（PSD）
        self.L_net = nn.Sequential(
            nn.Linear(dim, hidden), nn.Tanh(),
            nn.Linear(hidden, dim * dim)
        )
        self.dim = dim

    def forward(self, z):
        z = z.requires_grad_(True)
        H = self.H_net(z).sum()
        dH = torch.autograd.grad(H, z, create_graph=True)[0]
        # 标准辛矩阵 J（canonical form）
        n = self.dim // 2
        J = torch.zeros(self.dim, self.dim)
        J[:n, n:] = torch.eye(n)
        J[n:, :n] = -torch.eye(n)
        # 学习得到的耗散矩阵 R = L @ L^T
        L = self.L_net(z).reshape(-1, self.dim, self.dim)
        R = torch.bmm(L, L.transpose(1, 2))
        # 动力学方程：dz/dt = (J - R) @ dH/dz
        JmR = J.unsqueeze(0) - R
        return torch.bmm(JmR, dH.unsqueeze(-1)).squeeze(-1)

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import numpy as np

def three_body_hamiltonian(state):
    # state = [x1,y1, x2,y2, x3,y3, px1,py1, px2,py2, px3,py3]
    # 哈密顿量 H = ∑(|p_i|² / (2m_i)) − ∑_{i<j} G·m_i·m_j / |r_i − r_j|
    q = state[:6].reshape(3, 2)
    p = state[6:].reshape(3, 2)
    m = np.array([1.0, 1.0, 1.0])  # 质量相等
    G = 1.0
    T = sum(np.sum(p[i]**2) / (2 * m[i]) for i in range(3))
    V = 0
    for i in range(3):
        for j in range(i+1, 3):
            r = np.linalg.norm(q[i] - q[j])
            V -= G * m[i] * m[j] / max(r, 1e-6)
    return T + V

def leapfrog_3body(state, dt, n_steps):
    # 三体问题的辛积分器（蛙跳法）
    q = state[:6].reshape(3, 2).copy()
    p = state[6:].reshape(3, 2).copy()
    m = np.array([1.0, 1.0, 1.0])
    G = 1.0

    def grad_V(q):
        # 势能梯度的负值：−∂V/∂q_i = ∑_{j≠i} G·m_i·m_j·(q_j − q_i) / |r_{ij}|³
        force = np.zeros_like(q)
        for i in range(3):
            for j in range(3):
                if i != j:
                    r = q[j] - q[i]
                    dist = max(np.linalg.norm(r), 1e-6)
                    force[i] += G * m[i] * m[j] * r / dist**3
        return force

    traj = [np.concatenate([q.ravel(), p.ravel()])]
    for _ in range(n_steps):
        p += 0.5 * dt * grad_V(q)  # 半步动量更新（“半踢”）
        for i in range(3):
            q[i] += dt * p[i] / m[i]  # 全步位置更新（“漂移”）
        p += 0.5 * dt * grad_V(q)  # 半步动量更新（“半踢”）
        traj.append(np.concatenate([q.ravel(), p.ravel()]))

    return np.array(traj)

q0 = np.array([0.97, -0.24, -0.97, 0.24, 0.0, 0.0])
p0 = np.array([0.47, 0.24, 0.47, 0.24, -0.94, -0.48])
state0 = np.concatenate([q0, p0])

traj = leapfrog_3body(state0, dt=0.001, n_steps=50000)
H0 = three_body_hamiltonian(traj[0])
H_final = three_body_hamiltonian(traj[-1])
print(f"50 秒内的能量漂移：{abs(H_final - H0) / abs(H0) * 100:.6f}%")