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hpi/nn_models.py

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+"""Neural network models for HPI: CriticNN, ActorNN, and cost function.
+
+Replaces the polynomial basis functions (symbolic_engine.py) with
+data-adaptive NN representations that avoid multicollinearity issues.
+"""
+
+import torch
+import torch.nn as nn
+
+
+class CriticNN(nn.Module):
+    """Value function approximator V(x) >= 0 with structural V(0)=0.
+
+    Architecture:
+        Linear(2, 32, bias=False) -> Tanh
+        Linear(32, 32, bias=False) -> Tanh
+        Linear(32, 2, bias=False)          # two raw outputs z1, z2
+        V(x) = z1(x)^2 + z2(x)^2           # guaranteed V>=0 and V(0)=0
+
+    bias=False everywhere ensures V(0) = 0 structurally.
+    Two output heads allow the Hessian at the origin to be rank-2 (strict PD possible).
+    """
+
+    def __init__(self):
+        super().__init__()
+        self.fc1 = nn.Linear(2, 32, bias=False)
+        self.fc2 = nn.Linear(32, 32, bias=False)
+        self.fc3 = nn.Linear(32, 2, bias=False)
+
+    def forward(self, x):
+        z = torch.tanh(self.fc1(x))
+        z = torch.tanh(self.fc2(z))
+        z = self.fc3(z)  # (..., 2): [z1, z2]
+        return (z[:, 0:1] ** 2 + z[:, 1:2] ** 2).squeeze(-1)  # (...,) scalar
+
+    def get_raw_z(self, x):
+        """Return the two raw outputs z1, z2 before squaring."""
+        z = torch.tanh(self.fc1(x))
+        z = torch.tanh(self.fc2(z))
+        return self.fc3(z)  # (..., 2)
+
+
+class ActorNN(nn.Module):
+    """Policy network u(x): R^2 -> R.
+
+    Architecture:
+        Linear(2, 32, bias=True) -> Tanh
+        Linear(32, 32, bias=True) -> Tanh
+        Linear(32, 1, bias=True)            # signed control output
+    """
+
+    def __init__(self):
+        super().__init__()
+        self.fc1 = nn.Linear(2, 32, bias=True)
+        self.fc2 = nn.Linear(32, 32, bias=True)
+        self.fc3 = nn.Linear(32, 1, bias=True)
+
+    def forward(self, x):
+        h = torch.tanh(self.fc1(x))
+        h = torch.tanh(self.fc2(h))
+        return self.fc3(h).squeeze(-1)  # (...,) scalar
+
+
+def compute_q(x):
+    """Cost function Q(x) = x1^2 + x2^2.
+
+    Args:
+        x: torch.Tensor of shape (..., 2).
+
+    Returns:
+        torch.Tensor of shape (...,).
+    """
+    return x[..., 0] ** 2 + x[..., 1] ** 2
+
+
+def check_positive_definite(critic_nn, X_traj=None):
+    """Check if the critic NN corresponds to a positive-definite value function.
+
+    Two checks:
+    1. Hessian at origin is strictly positive definite (necessary condition).
+    2. V(x) > 0 on trajectory data (sanity check, structural guarantee makes
+       this always true, but verifies the NN hasn't degenerated).
+
+    Args:
+        critic_nn: CriticNN instance.
+        X_traj: Optional trajectory data (M, 2) numpy array for additional check.
+
+    Returns:
+        bool: True if all checks pass.
+    """
+    # 1. Hessian at origin
+    x0 = torch.zeros(1, 2, requires_grad=True)
+    V0 = critic_nn(x0)
+    grad_V = torch.autograd.grad(V0.sum(), x0, create_graph=True)[0]  # (1, 2)
+
+    h11 = torch.autograd.grad(grad_V[0, 0], x0, retain_graph=True)[0][0, 0].item()
+    h22 = torch.autograd.grad(grad_V[0, 1], x0, retain_graph=True)[0][0, 1].item()
+    h12 = torch.autograd.grad(grad_V[0, 0], x0, retain_graph=True)[0][0, 1].item()
+
+    hessian_pd = bool(h11 > 1e-6 and h22 > 1e-6 and h11 * h22 - h12 ** 2 > 1e-12)
+
+    if not hessian_pd:
+        return False
+
+    # 2. Optional: verify V(x) is well-behaved on trajectory data
+    if X_traj is not None:
+        x_t = torch.tensor(X_traj, dtype=torch.float32)
+        with torch.no_grad():
+            V_traj = critic_nn(x_t)
+        if torch.any(torch.isnan(V_traj)) or torch.any(torch.isinf(V_traj)):
+            return False
+        # V should be finite and non-negative everywhere on trajectory
+        if torch.any(V_traj < -1e-4):
+            return False
+
+    return True
+
+
+def check_lyapunov_decrease(critic_nn, X, f_X):
+    """Check if V satisfies the Lyapunov decrease condition on trajectory data.
+
+    Verifies: dV/dx · f(x) < 0 for states away from origin.
+    This is a necessary condition for V to be a valid Lyapunov function.
+
+    Args:
+        critic_nn: CriticNN instance.
+        X: State trajectory (M, 2) numpy array.
+        f_X: Drift dynamics f(X) evaluated on trajectory, (M, 2) numpy array.
+
+    Returns:
+        float: Fraction of points (away from origin) where V_dot < 0.
+    """
+    x_t = torch.tensor(X, dtype=torch.float32, requires_grad=True)
+    f_t = torch.tensor(f_X, dtype=torch.float32)
+    V = critic_nn(x_t)
+    grad_V = torch.autograd.grad(V.sum(), x_t, create_graph=False)[0]
+    V_dot = (grad_V * f_t).sum(dim=1)  # (M,)
+
+    # Only check points away from origin (||x|| > 0.05)
+    norms = torch.norm(x_t, dim=1)
+    mask = norms > 0.05
+    if mask.sum() == 0:
+        return 1.0
+
+    V_dot_filtered = V_dot[mask]
+    fraction_negative = (V_dot_filtered < 0).float().mean().item()
+    return fraction_negative