diff --git a/src/moscot/base/problems/problem.py b/src/moscot/base/problems/problem.py
index c602e1a22..e2b47cabb 100644
--- a/src/moscot/base/problems/problem.py
+++ b/src/moscot/base/problems/problem.py
@@ -403,6 +403,21 @@ def solve(
         """
         solver_class = backends.get_solver(self.problem_kind, backend=backend, return_class=True)
         init_kwargs, call_kwargs = solver_class._partition_kwargs(**kwargs)
+        # if linear problem, then alpha is 0.0 by default
+        # if quadratic problem, then alpha is 1.0 by default
+        alpha = call_kwargs.get("alpha", 0.0 if self.problem_kind == "linear" else 1.0)
+        if alpha < 0.0 or alpha > 1.0:
+            raise ValueError("Expected `alpha` to be in the range `[0, 1]`, found `{alpha}`.")
+        if self.problem_kind == "linear" and (alpha != 0.0 or not (self.x is None or self.y is None)):
+            raise ValueError("Unable to solve a linear problem with `alpha != 0` or `x` and `y` supplied.")
+        if self.problem_kind == "quadratic":
+            if self.x is None or self.y is None:
+                raise ValueError("Unable to solve a quadratic problem without `x` and `y` supplied.")
+            if alpha != 1.0 and self.xy is None:  # means FGW case
+                raise ValueError(
+                    "`alpha` must be 1.0 for quadratic problems without `xy` supplied. See `FGWProblem` class."
+                )
+
         self._solver = solver_class(**init_kwargs)
 
         self._solution = self._solver(  # type: ignore[misc]
diff --git a/src/moscot/problems/generic/_generic.py b/src/moscot/problems/generic/_generic.py
index 09f1e99ba..6f21cc964 100644
--- a/src/moscot/problems/generic/_generic.py
+++ b/src/moscot/problems/generic/_generic.py
@@ -593,7 +593,7 @@ def prepare(
 
     def solve(
         self,
-        alpha: float = 1.0,
+        alpha: float = 0.5,
         epsilon: float = 1e-3,
         tau_a: float = 1.0,
         tau_b: float = 1.0,
@@ -622,7 +622,7 @@ def solve(
         Parameters
         ----------
         alpha
-            Parameter in :math:`(0, 1]` that interpolates between the :term:`quadratic term` and
+            Parameter in :math:`(0, 1)` that interpolates between the :term:`quadratic term` and
             the :term:`linear term`. :math:`\alpha = 1` corresponds to the pure :term:`Gromov-Wasserstein` problem while
             :math:`\alpha \to 0` corresponds to the pure :term:`linear problem`.
         epsilon
@@ -672,6 +672,8 @@ def solve(
         - :attr:`solutions` - the :term:`OT` solutions for each subproblem.
         - :attr:`stage` - set to ``'solved'``.
         """
+        if alpha == 1.0:
+            raise ValueError("The `FGWProblem` is equivalent to the `GWProblem` when `alpha=1.0`.")
         return CompoundProblem.solve(
             self,  # type: ignore[return-value, arg-type]
             alpha=alpha,
diff --git a/tests/problems/cross_modality/test_translation_problem.py b/tests/problems/cross_modality/test_translation_problem.py
index f2c3d0314..a90845d75 100644
--- a/tests/problems/cross_modality/test_translation_problem.py
+++ b/tests/problems/cross_modality/test_translation_problem.py
@@ -83,7 +83,14 @@ def test_prepare_external_star_policy(
 
     @pytest.mark.parametrize(
         ("epsilon", "alpha", "rank", "initializer"),
-        [(1e-2, 0.9, -1, None), (2, 0.5, -1, "random"), (2, 0.5, -1, "rank2"), (2, 0.1, -1, None)],
+        [
+            (1e-2, 0.9, -1, None),
+            (2, 0.5, -1, "random"),
+            (2, 1.0, -1, "rank2"),
+            (2, 0.1, -1, None),
+            (2, 1.0, -1, None),
+            (1.3, 1.0, -1, "random"),
+        ],
     )
     @pytest.mark.parametrize("src_attr", ["emb_src", {"attr": "obsm", "key": "emb_src"}])
     @pytest.mark.parametrize("tgt_attr", ["emb_tgt", {"attr": "obsm", "key": "emb_tgt"}])
@@ -104,7 +111,8 @@ def test_solve_balanced(
             kwargs["initializer"] = initializer
 
         tp = TranslationProblem(adata_src, adata_tgt)
-        tp = tp.prepare(batch_key="batch", src_attr=src_attr, tgt_attr=tgt_attr)
+        joint_attr = None if alpha == 1.0 else {"attr": "obsm", "key": "X_pca"}
+        tp = tp.prepare(batch_key="batch", src_attr=src_attr, tgt_attr=tgt_attr, joint_attr=joint_attr)
         tp = tp.solve(epsilon=epsilon, alpha=alpha, rank=rank, **kwargs)
 
         for key, subsol in tp.solutions.items():
diff --git a/tests/problems/space/test_mapping_problem.py b/tests/problems/space/test_mapping_problem.py
index 6a8020a4f..65e5641ee 100644
--- a/tests/problems/space/test_mapping_problem.py
+++ b/tests/problems/space/test_mapping_problem.py
@@ -120,6 +120,7 @@ def test_solve_balanced(
                 return  # TODO(@MUCDK) fix after refactoring
         mp = MappingProblem(adataref, adatasp)
         mp = mp.prepare(batch_key="batch", sc_attr=sc_attr, var_names=var_names)
+        alpha = alpha if mp.filtered_vars is not None else 1.0
         mp = mp.solve(epsilon=epsilon, alpha=alpha, rank=rank, **kwargs)
 
         for prob_key in mp: