diff --git a/deepxde/data/pde.py b/deepxde/data/pde.py
index 67f900648..72be31514 100644
--- a/deepxde/data/pde.py
+++ b/deepxde/data/pde.py
@@ -4,7 +4,12 @@
 from .. import backend as bkd
 from .. import config
 from ..backend import backend_name
-from ..utils import get_num_args, run_if_all_none, mpi_scatter_from_rank0
+from ..utils import (
+    get_num_args,
+    has_default_values,
+    run_if_all_none,
+    mpi_scatter_from_rank0,
+)
 
 
 class PDE(Data):
@@ -150,9 +155,18 @@ def losses(self, targets, outputs, loss_fn, inputs, model, aux=None):
             elif get_num_args(self.pde) == 3:
                 if self.auxiliary_var_fn is not None:
                     f = self.pde(inputs, outputs_pde, model.net.auxiliary_vars)
-                elif backend_name == "jax" and len(aux) == 2:
+                elif backend_name == "jax":
                     # JAX inverse problem requires unknowns as the input.
-                    f = self.pde(inputs, outputs_pde, unknowns=aux[1])
+                    if len(aux) == 2:
+                        # External trainable variables in aux[1] are used for unknowns
+                        f = self.pde(inputs, outputs_pde, unknowns=aux[1])
+                    elif len(aux) == 1 and has_default_values(self.pde)[-1]:
+                        # No external trainable variables, default values are used for unknowns
+                        f = self.pde(inputs, outputs_pde)
+                    else:
+                        raise ValueError(
+                            "Default unknowns are required if no trainable variables are provided."
+                        )
                 else:
                     raise ValueError("Auxiliary variable function not defined.")
             if not isinstance(f, (list, tuple)):
diff --git a/deepxde/model.py b/deepxde/model.py
index 1644c2529..e1d898506 100644
--- a/deepxde/model.py
+++ b/deepxde/model.py
@@ -401,10 +401,10 @@ def _compile_jax(self, lr, loss_fn, decay):
         if self.params is None:
             key = jax.random.PRNGKey(config.jax_random_seed)
             self.net.params = self.net.init(key, self.data.test()[0])
-            external_trainable_variables_arr = [
-                var.value for var in self.external_trainable_variables
-            ]
-            self.params = [self.net.params, external_trainable_variables_arr]
+        external_trainable_variables_val = [
+            var.value for var in self.external_trainable_variables
+        ]
+        self.params = [self.net.params, external_trainable_variables_val]
         # TODO: learning rate decay
         self.opt = optimizers.get(self.opt_name, learning_rate=lr)
         self.opt_state = self.opt.init(self.params)
diff --git a/deepxde/utils/internal.py b/deepxde/utils/internal.py
index 1b57f9d0d..dfa1a55f0 100644
--- a/deepxde/utils/internal.py
+++ b/deepxde/utils/internal.py
@@ -202,6 +202,18 @@ def get_num_args(func):
     params = inspect.signature(func).parameters
     return len(params) - ("self" in params)
 
+def has_default_values(func):
+    """
+    Check if the given function has default values for its parameters.
+
+    Args:
+        func (function): The function to inspect.
+
+    Returns:
+        list: A list of boolean values indicating whether each parameter has a default value.
+    """
+    params = inspect.signature(func).parameters.values()
+    return [param.default is not inspect.Parameter.empty for param in params]
 
 def mpi_scatter_from_rank0(array, drop_last=True):
     """Scatter the given array into continuous subarrays of equal size from rank 0 to all ranks.