diff --git a/ppsci/autodiff/ad.py b/ppsci/autodiff/ad.py
index b50650675..a2a7897b1 100644
--- a/ppsci/autodiff/ad.py
+++ b/ppsci/autodiff/ad.py
@@ -21,7 +21,7 @@
 import paddle
 
 
-class Jacobian:
+class _Jacobian:
     """Compute Jacobian matrix J: J[i][j] = dy_i/dx_j, where i = 0, ..., dim_y-1 and
     j = 0, ..., dim_x - 1.
 
@@ -47,19 +47,23 @@ def __call__(self, i: int = 0, j: Optional[int] = None) -> "paddle.Tensor":
         J[i].
         """
         if not 0 <= i < self.dim_y:
-            raise ValueError(f"i={i} is not valid.")
+            raise ValueError(f"i({i}) should in range [0, {self.dim_y}).")
         if j is not None and not 0 <= j < self.dim_x:
-            raise ValueError(f"j={j} is not valid.")
+            raise ValueError(f"j({j}) should in range [0, {self.dim_x}).")
         # Compute J[i]
         if i not in self.J:
             y = self.ys[:, i : i + 1] if self.dim_y > 1 else self.ys
             self.J[i] = paddle.grad(y, self.xs, create_graph=True)[0]
 
-        return self.J[i] if j is None or self.dim_x == 1 else self.J[i][:, j : j + 1]
+        return self.J[i] if (j is None or self.dim_x == 1) else self.J[i][:, j : j + 1]
 
 
 class Jacobians:
-    """Compute multiple Jacobians.
+    r"""Compute multiple Jacobians.
+
+    $$
+    \rm Jacobian(ys, xs, i, j) = \dfrac{\partial ys_i}{\partial xs_j}
+    $$
 
     A new instance will be created for a new pair of (output, input). For the (output,
     input) pair that has been computed before, it will reuse the previous instance,
@@ -97,21 +101,25 @@ def __call__(
         """
         key = (ys, xs)
         if key not in self.Js:
-            self.Js[key] = Jacobian(ys, xs)
+            self.Js[key] = _Jacobian(ys, xs)
         return self.Js[key](i, j)
 
-    def clear(self):
+    def _clear(self):
         """Clear cached Jacobians."""
         self.Js = {}
 
 
-class Hessian:
+# Use high-order differentiation with singleton pattern for convenient
+jacobian = Jacobians()
+
+
+class _Hessian:
     """Compute Hessian matrix H: H[i][j] = d^2y / dx_i dx_j, where i,j = 0,..., dim_x-1.
 
     It is lazy evaluation, i.e., it only computes H[i][j] when needed.
 
     Args:
-        y: Output Tensor of shape (batch_size, 1) or (batch_size, dim_y > 1).
+        ys: Output Tensor of shape (batch_size, 1) or (batch_size, dim_y > 1).
         xs: Input Tensor of shape (batch_size, dim_x).
         component: If `y` has the shape (batch_size, dim_y > 1), then `y[:, component]`
             is used to compute the Hessian. Do not use if `y` has the shape (batch_size,
@@ -122,29 +130,32 @@ class Hessian:
 
     def __init__(
         self,
-        y: "paddle.Tensor",
+        ys: "paddle.Tensor",
         xs: "paddle.Tensor",
         component: Optional[int] = None,
         grad_y: Optional["paddle.Tensor"] = None,
     ):
-        dim_y = y.shape[1]
+        dim_y = ys.shape[1]
 
         if dim_y > 1:
             if component is None:
-                raise ValueError("The component of y is missing.")
+                raise ValueError(
+                    f"component({component}) can not be None when dim_y({dim_y})>1."
+                )
             if component >= dim_y:
                 raise ValueError(
-                    f"The component of y={component} cannot be larger than the "
-                    f"dimension={dim_y}."
+                    f"component({component}) should be smaller than dim_y({dim_y})."
                 )
         else:
             if component is not None:
-                raise ValueError("Do not use component for 1D y.")
+                raise ValueError(
+                    f"component{component} should be set to None when dim_y({dim_y})=1."
+                )
             component = 0
 
         if grad_y is None:
-            grad_y = jacobian(y, xs, i=component, j=None)
-        self.H = Jacobian(grad_y, xs)
+            grad_y = jacobian(ys, xs, i=component, j=None)
+        self.H = _Jacobian(grad_y, xs)
 
     def __call__(self, i: int = 0, j: int = 0):
         """Returns H[`i`][`j`]."""
@@ -152,7 +163,11 @@ def __call__(self, i: int = 0, j: int = 0):
 
 
 class Hessians:
-    """Compute multiple Hessians.
+    r"""Compute multiple Hessians.
+
+    $$
+    \rm Hessian(ys, xs, component, i, j) = \dfrac{\partial ys_{component}}{\partial xs_i \partial xs_j}
+    $$
 
     A new instance will be created for a new pair of (output, input). For the (output,
     input) pair that has been computed before, it will reuse the previous instance,
@@ -197,20 +212,19 @@ def __call__(
         """
         key = (ys, xs, component)
         if key not in self.Hs:
-            self.Hs[key] = Hessian(ys, xs, component=component, grad_y=grad_y)
+            self.Hs[key] = _Hessian(ys, xs, component=component, grad_y=grad_y)
         return self.Hs[key](i, j)
 
-    def clear(self):
+    def _clear(self):
         """Clear cached Hessians."""
         self.Hs = {}
 
 
 # Use high-order differentiation with singleton pattern for convenient
-jacobian = Jacobians()
 hessian = Hessians()
 
 
 def clear():
     """Clear cached Jacobians and Hessians."""
-    jacobian.clear()
-    hessian.clear()
+    jacobian._clear()
+    hessian._clear()