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diff --git a/test/test_weak.py b/test/test_weak.py
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+"""Tests for main module ot.weak """
+
+# Author: Remi Flamary <remi.flamary@unice.fr>
+#
+# License: MIT License
+
+import ot
+import numpy as np
+
+
+def test_weak_ot():
+    # test weak ot solver and identity stationary point
+    n = 50
+    rng = np.random.RandomState(0)
+
+    xs = rng.randn(n, 2)
+    xt = rng.randn(n, 2)
+    u = ot.utils.unif(n)
+
+    G, log = ot.weak_optimal_transport(xs, xt, u, u, log=True)
+
+    # check constraints
+    np.testing.assert_allclose(u, G.sum(1))
+    np.testing.assert_allclose(u, G.sum(0))
+
+    # chaeck that identity is recovered
+    G = ot.weak_optimal_transport(xs, xs, G0=np.eye(n) / n)
+
+    # check G is identity
+    np.testing.assert_allclose(G, np.eye(n) / n)
+
+    # check constraints
+    np.testing.assert_allclose(u, G.sum(1))
+    np.testing.assert_allclose(u, G.sum(0))
+
+
+def test_weak_ot_bakends(nx):
+    # test weak ot solver for different backends
+    n = 50
+    rng = np.random.RandomState(0)
+
+    xs = rng.randn(n, 2)
+    xt = rng.randn(n, 2)
+    u = ot.utils.unif(n)
+
+    G = ot.weak_optimal_transport(xs, xt, u, u)
+
+    xs2, xt2, u2 = nx.from_numpy(xs, xt, u)
+
+    G2 = ot.weak_optimal_transport(xs2, xt2, u2, u2)
+
+    np.testing.assert_allclose(nx.to_numpy(G2), G)