diff --git a/optax/_src/transform.py b/optax/_src/transform.py
index 70f30d817..2e9de4212 100644
--- a/optax/_src/transform.py
+++ b/optax/_src/transform.py
@@ -15,7 +15,7 @@
 """Gradient transformations."""
 
 import functools
-from typing import NamedTuple, Optional, Union
+from typing import NamedTuple, Optional
 
 import chex
 import jax
@@ -44,7 +44,8 @@ class ScaleByRssState(NamedTuple):
 
 
 def scale_by_rss(
-    initial_accumulator_value: float = 0.1, eps: float = 1e-7
+    initial_accumulator_value: float | jax.Array = 0.1,
+    eps: float | jax.Array = 1e-7,
 ) -> base.GradientTransformation:
   """Rescale updates by the root of the sum of all squared gradients to date.
 
@@ -93,9 +94,9 @@ class ScaleByRmsWithCountState(NamedTuple):
 
 
 def scale_by_rms(
-    decay: float = 0.9,
-    eps: float = 1e-8,
-    initial_scale: float = 0.0,
+    decay: float | jax.Array = 0.9,
+    eps: float | jax.Array = 1e-8,
+    initial_scale: float | jax.Array = 0.0,
     eps_in_sqrt: bool = True,
     bias_correction: bool = False,
 ) -> base.GradientTransformation:
@@ -168,9 +169,9 @@ class ScaleByRStdDevWithCountState(NamedTuple):
 
 
 def scale_by_stddev(
-    decay: float = 0.9,
-    eps: float = 1e-8,
-    initial_scale: float = 0.0,
+    decay: float | jax.Array = 0.9,
+    eps: float | jax.Array = 1e-8,
+    initial_scale: float | jax.Array = 0.0,
     eps_in_sqrt: bool = True,
     bias_correction: bool = False,
 ) -> base.GradientTransformation:
@@ -244,10 +245,10 @@ class ScaleByAdamState(NamedTuple):
 
 
 def scale_by_adam(
-    b1: float = 0.9,
-    b2: float = 0.999,
-    eps: float = 1e-8,
-    eps_root: float = 0.0,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-8,
+    eps_root: float | jax.Array = 0.0,
     mu_dtype: Optional[chex.ArrayDType] = None,
     *,
     nesterov: bool = False,
@@ -317,10 +318,10 @@ class ScaleByAmsgradState(NamedTuple):
 
 
 def scale_by_amsgrad(
-    b1: float = 0.9,
-    b2: float = 0.999,
-    eps: float = 1e-8,
-    eps_root: float = 0.0,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-8,
+    eps_root: float | jax.Array = 0.0,
     mu_dtype: Optional[chex.ArrayDType] = None,
 ) -> base.GradientTransformation:
   """Rescale updates according to the AMSGrad algorithm.
@@ -373,7 +374,9 @@ def update_fn(updates, state, params=None):
 
 
 def scale_by_adamax(
-    b1: float = 0.9, b2: float = 0.999, eps: float = 1e-8
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-8,
 ) -> base.GradientTransformation:
   """Rescale updates according to the Adamax algorithm.
 
@@ -414,8 +417,8 @@ class ScaleByLionState(NamedTuple):
 
 
 def scale_by_lion(
-    b1: float = 0.9,
-    b2: float = 0.99,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.99,
     mu_dtype: Optional[chex.ArrayDType] = None,
 ) -> base.GradientTransformation:
   """Rescale updates according to the Lion algorithm.
@@ -451,7 +454,9 @@ def update_fn(updates, state, params=None):
   return base.GradientTransformation(init_fn, update_fn)
 
 
-def scale(step_size: float) -> base.GradientTransformation:
+def scale(
+    step_size: float | jax.Array
+) -> base.GradientTransformation:
   """Scale updates by some fixed scalar `step_size`.
 
   Args:
@@ -470,7 +475,7 @@ def update_fn(updates, state, params=None):
 
 
 def scale_by_param_block_norm(
-    min_scale: float = 1e-3,
+    min_scale: float | jax.Array = 1e-3
 ) -> base.GradientTransformation:
   """Scale updates for each param block by the norm of that block's parameters.
 
@@ -496,7 +501,7 @@ def update_fn(updates, state, params):
 
 
 def scale_by_param_block_rms(
-    min_scale: float = 1e-3,
+    min_scale: float | jax.Array = 1e-3,
 ) -> base.GradientTransformation:
   """Scale updates by rms of the gradient for each param vector or matrix.
 
@@ -656,10 +661,10 @@ class ScaleByBeliefState(NamedTuple):
 
 
 def scale_by_belief(
-    b1: float = 0.9,
-    b2: float = 0.999,
-    eps: float = 1e-16,
-    eps_root: float = 1e-16,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-16,
+    eps_root: float | jax.Array = 1e-16,
     *,
     nesterov: bool = False,
 ) -> base.GradientTransformation:
@@ -713,11 +718,11 @@ def update_fn(updates, state, params=None):
 
 
 def scale_by_yogi(
-    b1: float = 0.9,
-    b2: float = 0.999,
-    eps: float = 1e-3,
-    eps_root: float = 0.0,
-    initial_accumulator_value: float = 1e-6,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-3,
+    eps_root: float | jax.Array = 0.0,
+    initial_accumulator_value: float | jax.Array = 1e-6,
 ) -> base.GradientTransformation:
   """Rescale updates according to the Yogi algorithm.
 
@@ -767,11 +772,11 @@ def update_fn(updates, state, params=None):
 
 
 def scale_by_radam(
-    b1: float = 0.9,
-    b2: float = 0.999,
-    eps: float = 1e-8,
-    eps_root: float = 0.0,
-    threshold: float = 5.0,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.999,
+    eps: float | jax.Array = 1e-8,
+    eps_root: float | jax.Array = 0.0,
+    threshold: float | jax.Array = 5.0,
     *,
     nesterov: bool = False,
 ) -> base.GradientTransformation:
@@ -991,9 +996,9 @@ def update_fn(updates, state, params=None):
 
 
 def scale_by_trust_ratio(
-    min_norm: float = 0.0,
-    trust_coefficient: float = 1.0,
-    eps: float = 0.0,
+    min_norm: float | jax.Array = 0.0,
+    trust_coefficient: float | jax.Array = 1.0,
+    eps: float | jax.Array = 0.0,
 ) -> base.GradientTransformation:
   """Scale updates by `trust ratio`.
 
@@ -1121,7 +1126,9 @@ class ScaleBySM3State(NamedTuple):
 
 
 def scale_by_sm3(
-    b1: float = 0.9, b2: float = 1.0, eps: float = 1e-8
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 1.0,
+    eps: float | jax.Array = 1e-8,
 ) -> base.GradientTransformation:
   """Scale updates by `sm3`.
 
@@ -1198,11 +1205,11 @@ class ScaleByNovogradState(NamedTuple):
 
 
 def scale_by_novograd(
-    b1: float = 0.9,
-    b2: float = 0.25,
-    eps: float = 1e-8,
-    eps_root: float = 0.0,
-    weight_decay: float = 0.0,
+    b1: float | jax.Array = 0.9,
+    b2: float | jax.Array = 0.25,
+    eps: float | jax.Array = 1e-8,
+    eps_root: float | jax.Array = 0.0,
+    weight_decay: float | jax.Array = 0.0,
     mu_dtype: Optional[chex.ArrayDType] = None,
 ) -> base.GradientTransformation:
   """Computes NovoGrad updates.
@@ -1277,7 +1284,8 @@ class ScaleByOptimisticGradientState(NamedTuple):
 
 
 def scale_by_optimistic_gradient(
-    alpha: float = 1.0, beta: float = 1.0
+    alpha: float | jax.Array = 1.0,
+    beta: float | jax.Array = 1.0,
 ) -> base.GradientTransformation:
   """Compute generalized optimistic gradients.
 
@@ -1494,8 +1502,8 @@ def _precondition_by_lbfgs(
     diff_params_memory: chex.ArrayTree,
     diff_updates_memory: chex.ArrayTree,
     weights_memory: chex.Array,
-    identity_scale: Union[float, jax.Array],
-    memory_idx: Union[int, jax.Array],
+    identity_scale: float | jax.Array,
+    memory_idx: int | jax.Array,
 ) -> base.Updates:
   r"""Multiplies updates by an approximation of the inverse Hessian.