diff --git a/swirl_dynamics/projects/debiasing/rectified_flow/models.py b/swirl_dynamics/projects/debiasing/rectified_flow/models.py
index bb18fdc..62b0486 100644
--- a/swirl_dynamics/projects/debiasing/rectified_flow/models.py
+++ b/swirl_dynamics/projects/debiasing/rectified_flow/models.py
@@ -106,6 +106,8 @@ class ReFlowModel(models.BaseModel):
       This should be close to 1.
     num_eval_time_levels: Number of times at which the flow will be sampled for
       each trajectory between x_0 and x_1.
+    weighted_norm: The norm to use for the loss, if None we use euclidean norm,
+      otherwise we use weighted norm.
   """
 
   input_shape: tuple[int, ...]
@@ -122,10 +124,23 @@ class ReFlowModel(models.BaseModel):
   max_eval_time_lvl: float = 1.0 - 1e-4  # It should be close to 1.
   num_eval_time_levels: ClassVar[int] = 10
 
+  weighted_norm: Array | None = None
+
   def initialize(self, rng: Array):
     # TODO: Add a dtype object to ensure consistency of types.
     x = jnp.ones((1,) + self.input_shape)
 
+    # If weighted norm is provided, it must have the right shape.
+    if self.weighted_norm is not None:
+      if (
+          self.weighted_norm.shape[0] != 1
+          or self.weighted_norm.shape[1:] != x.shape[1:]
+      ):
+        raise ValueError(
+            "Weighted norm shape must be (1, *x.shape[1:]) instead we have"
+            f" {self.weighted_norm.shape}, with x.shape = {x.shape}"
+        )
+
     return self.flow_model.init(
         rng, x=x, sigma=jnp.ones((1,)), is_training=False
     )
@@ -172,8 +187,16 @@ def loss_fn(
         is_training=True,
         rngs={"dropout": dropout_rng},
     )
-    # Eq. (1) in [1].
-    loss = jnp.mean(jnp.square((batch["x_1"] - batch["x_0"]) - v_t))
+
+    # TODO: Define a metric class that incorporates the weights.
+    if self.weighted_norm is None:
+      weighted_norm = 1.0
+    else:
+      weighted_norm = self.weighted_norm
+
+    # Eq. (1) in [1], but with a possibly weighted norm.
+    error = ((batch["x_1"] - batch["x_0"]) - v_t)
+    loss = jnp.mean(weighted_norm * jnp.square(error))
     metric = dict(loss=loss)
     return loss, (metric, mutables)