diff --git a/descent/targets/thermo.py b/descent/targets/thermo.py
index f3e808b..3c52b2f 100644
--- a/descent/targets/thermo.py
+++ b/descent/targets/thermo.py
@@ -1,7 +1,9 @@
 """Train against thermodynamic properties."""
 import contextlib
 import hashlib
+import logging
 import pathlib
+import pickle
 import typing
 
 import numpy
@@ -13,6 +15,9 @@
 
 import descent.utils.molecule
 
+_LOGGER = logging.getLogger(__name__)
+
+
 DataType = typing.Literal["density", "hvap", "hmix"]
 
 DATA_TYPES = typing.get_args(DataType)
@@ -235,7 +240,7 @@ def _bulk_config(temperature: float, pressure: float) -> SimulationConfig:
             n_steps=500000,
             timestep=2.0 * openmm.unit.femtosecond,
         ),
-        production_frequency=500,
+        production_frequency=1000,
     )
 
 
@@ -396,7 +401,7 @@ def _compute_averages(
     output_dir: pathlib.Path,
     cached_dir: pathlib.Path | None,
 ) -> dict[str, torch.Tensor]:
-    traj_hash = hashlib.sha256(key, usedforsecurity=False).hexdigest()
+    traj_hash = hashlib.sha256(pickle.dumps(key)).hexdigest()
     traj_name = f"{phase}-{traj_hash}-frames.msgpack"
 
     cached_path = None if cached_dir is None else cached_dir / traj_name
@@ -410,6 +415,9 @@ def _compute_averages(
                 system, force_field, cached_path, temperature, pressure
             )
 
+    if cached_path is not None:
+        _LOGGER.debug(f"unable to re-weight {key}: data exists={cached_path.exists()}")
+
     output_path = output_dir / traj_name
 
     config = default_config(phase, key.temperature, key.pressure)
@@ -548,4 +556,7 @@ def predict(
             torch.tensor(entry["value"]) * per_type_scales.get(entry["type"], 1.0)
         )
 
-    return torch.stack(reference), torch.stack(predicted)
+    predicted = torch.stack(predicted)
+    reference = torch.stack(reference).to(predicted.device)
+
+    return reference, predicted
diff --git a/descent/tests/utils/test_loss.py b/descent/tests/utils/test_loss.py
index 7b67a81..61f2451 100644
--- a/descent/tests/utils/test_loss.py
+++ b/descent/tests/utils/test_loss.py
@@ -1,6 +1,6 @@
 import torch
 
-from descent.utils.loss import to_closure
+from descent.utils.loss import approximate_hessian, to_closure
 
 
 def test_to_closure():
@@ -35,3 +35,21 @@ def mock_loss_fn(x: torch.Tensor, a: float, b: float) -> torch.Tensor:
     assert loss is not None
     assert grad is not None
     assert hess is None
+
+
+def test_approximate_hessian():
+    x = torch.tensor([1.0, 2.0, 3.0, 4.0], requires_grad=True)
+    y_pred = 5.0 * x**2 + 3.0 * x + 2.0
+
+    actual_hessian = approximate_hessian(x, y_pred)
+    expected_hess = torch.tensor(
+        [
+            [338.0, 0.0, 0.0, 0.0],
+            [0.0, 1058.0, 0.0, 0.0],
+            [0.0, 0.0, 2178.0, 0.0],
+            [0.0, 0.0, 0.0, 3698.0],
+        ]
+    )
+
+    assert actual_hessian.shape == expected_hess.shape
+    assert torch.allclose(actual_hessian, expected_hess)
diff --git a/descent/utils/loss.py b/descent/utils/loss.py
index 909ac70..b652d59 100644
--- a/descent/utils/loss.py
+++ b/descent/utils/loss.py
@@ -51,3 +51,23 @@ def closure_fn(
         return loss.detach(), gradient, hessian
 
     return closure_fn
+
+
+def approximate_hessian(x: torch.Tensor, y_pred: torch.Tensor):
+    """Compute the outer product approximation of the hessian of a least squares
+    loss function of the sum ``sum((y_pred - y_ref)**2)``.
+
+    Args:
+        x: The parameter tensor with ``shape=(n_parameters,)``.
+        y_pred: The values predicted using ``x`` with ``shape=(n_predications,)``.
+
+    Returns:
+        The outer product approximation of the hessian with ``shape=n_parameters
+    """
+
+    y_pred_grad = [torch.autograd.grad(y, x, retain_graph=True)[0] for y in y_pred]
+    y_pred_grad = torch.stack(y_pred_grad, dim=0)
+
+    return (
+        2.0 * torch.einsum("bi,bj->bij", y_pred_grad, y_pred_grad).sum(dim=0)
+    ).detach()